chapter · b : factor polynomials when the terms have a common factor, factoring out the greatest...
Post on 05-Jul-2020
2 Views
Preview:
TRANSCRIPT
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.1 Introduction to Factoring
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Find the greatest common factor, the GCF, of monomials.
b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.
c Factor certain expressions with four terms using factoring by grouping.
5.1 Introduction to Factoring
a Find the greatest common factor, the GCF, of monomials.
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The greatest of the common factors of several terms is called the greatest common factor, GCF.
EXAMPLE
5.1 Introduction to Factoring
a Find the greatest common factor, the GCF, of monomials.
2 Find the GCF of 180 and 420.
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.1 Introduction to Factoring
a Find the greatest common factor, the GCF, of monomials.
4 Find the GCF of 54, 90, and 252.
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.1 Introduction to Factoring
a Find the greatest common factor, the GCF, of monomials.
4
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring
a Find the greatest common factor, the GCF, of monomials.
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Consider the product
To factor the polynomial on the right, we reverse the process of multiplication:
5.1 Introduction to Factoring
Factor; Factorization
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To factor a polynomial is to express it as a product. A factor of a polynomial P is a polynomial that can be used to express P as a product. A factorization of a polynomial is an expression that names that polynomial as a product.
EXAMPLE
5.1 Introduction to Factoring
a Find the greatest common factor, the GCF, of monomials.
5 Find the GCF of and
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The greatest positive common factor of the coefficients is 3. Next, we find the GCF of the powers of x. That GCF is x2 because 2 is the smallest exponent of x. Thus the GCF of the set of monomials is 3x2.
5.1 Introduction to Factoring
To Find the GCF of Two or More Monomials
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1. Find the prime factorization of the coefficients, including –1 as a factor if any coefficient is negative.
2. Determine any common prime factors of the coefficients. For each one that occurs, include it as a factor of the GCF. If none occurs, use 1 as a factor.
3. Examine each of the variables as factors. If any appear as a factor of all the monomials, include it as a factor, using the smallest exponent of the variable. If none occurs in all the monomials, use 1 as a factor.
4. The GCF is the product of the results of steps (2) and (3).
5.1 Introduction to Factoring
b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To multiply a monomial and a polynomial with more than one term, we multiply each term of the polynomial by the monomial using the distributive laws:
To factor, we do the reverse. We express a polynomial as a product using the distributive laws in reverse:
5.1 Introduction to Factoring
b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.1 Introduction to Factoring
b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.
7 Factor:
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.1 Introduction to Factoring
b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.
8 Factor:
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.1 Introduction to Factoring
b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.
8
Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring
Tips for Factoring
Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
• Before doing any other kind of factoring, first try to factor out the GCF.
• Always check the result of factoring by multiplying.
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Certain polynomials with four terms can be factored using a method called factoring by grouping.
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Consider the four-term polynomial
There is no factor other than 1 that is common to all the terms. We can, however, factor separately:
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
Slide 21 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
This method of factoring is called factoring by grouping.
EXAMPLE
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
15 Factor by grouping.
Slide 22 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
15
Slide 23 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
We think through this process as follows:
EXAMPLE
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
17 Factor by grouping:
Slide 24 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
17
Slide 25 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.1 Introduction to Factoring
c Factor certain expressions with four terms using factoring by grouping.
17
Slide 26 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.2 Factoring Trinomials of the Type x2 + bx + c
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Compare the following multiplications:
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Note that for all four products: • The product of the two binomials is a trinomial. • The coefficient of x in the trinomial is the sum of the
constant terms in the binomials. • The constant term in the trinomial is the product of the
constant terms in the binomials.
These observations lead to a method for factoring certain trinomials.
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To factor x2 + 7x + 10, we think of FOIL in reverse. We multiplied x times x to get the first term of the trinomial, so we know that the first term of each binomial factor is x. Next, we look for numbers p and q such that
To get the middle term and the last term of the trinomial, we look for two numbers and whose product is 10 and whose sum is 7. Those numbers are 2 and 5. Thus the factorization is
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
1 Factor:
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.2 Factoring Trinomials of the Type x2 + bx + c
To Factor x2 + bx + c when c Is Positive
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
When the constant term of a trinomial is positive, look for two numbers with the same sign. The sign is that of the middle term:
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
2 Factor:
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
2
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Since the constant term, 12, is positive and the coefficient of the middle term, –8, is negative, we look for a factorization of 12 in which both factors are negative. Their sum must be –8.
EXAMPLE Solution
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
2
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The product of two binomials can have a negative constant term:
Note that when the signs of the constants in the binomials are reversed, only the sign of the middle term in the product changes.
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
3 Factor:
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The constant term, –20, must be expressed as the product of a negative number and a positive number. Since the sum of these two numbers must be negative (specifically, –8), the negative number must have the greater absolute value.
EXAMPLE Solution
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
3
Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.2 Factoring Trinomials of the Type x2 + bx + c
To Factor x2 + bx + c when c Is Negative
Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
When the constant term of a trinomial is negative, look for two numbers whose product is negative. One must be positive and the other negative:
Consider pairs of numbers for which the number with the larger absolute value has the same sign as b, the coefficient of the middle term.
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
4 Factor:
Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
4
Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
5 Factor:
Slide 21 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Consider this trinomial as We look for numbers p and q such that
The middle-term coefficient, –1, is small compared to –110. This tells us that the desired factors are close to each other in absolute value. The numbers we want are 10 and –11. The factorization is
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
6 Factor:
Slide 22 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
We consider the trinomial in the equivalent form
Think of –21b2 as the “constant” term and 4b as the “coefficient” of the middle term. Then try to express –21b2 as a product of two factors whose sum is 4b. Those factors are –3b and 7b.
The factorization is
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
7 Factor:
Slide 23 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
There are no factors whose sum is –1. Thus the polynomial is not factorable into factors that are polynomials with rational-number coefficients.
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 24 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
In this text, a polynomial like x2 – x + 5 that cannot be factored further is said to be prime. In more advanced courses, polynomials like x2 – x + 5 can be factored and are not considered prime.
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
Slide 25 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Often factoring requires two or more steps. In general, when told to factor, we should factor completely. This means that the final factorization should not contain any factors that can be factored further.
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
8 Factor:
Slide 26 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Always look first for a common factor. This time there is one, 2x.
EXAMPLE Solution
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
8
Slide 27 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Now consider
EXAMPLE Solution
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
8
Slide 28 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.2 Factoring Trinomials of the Type x2 + bx + c
To Factor x2 + bx + c
Slide 29 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1. First arrange in descending order. 2. Use a trial-and-error process that looks for factors of c
whose sum is b. 3. If c is positive, the signs of the factors are the same as
the sign of b. 4. If c is negative, one factor is positive and the other is
negative. If the sum of two factors is the opposite of b, changing the sign of each factor will give the desired factors whose sum is b.
5. Check by multiplying.
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
9 Factor:
Slide 30 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Write in descending order:
EXAMPLE
5.2 Factoring Trinomials of the Type x2 + bx + c
a Factor trinomials of the type x2 + bx + c by examining the constant term c.
9 Factor:
Slide 31 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Each of the following is also a correct answer:
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
We want to factor trinomials of the type ax2 + bx + c. Consider the following multiplication:
We reverse the above multiplication, using what we might call an “unFOIL” process.
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
We look for two binomials whose product is The answer is It turns out that is also a correct answer, but we generally choose an answer in which the first coefficients are positive.
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
The FOIL Method
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
The FOIL Method
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
1 Factor:
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1) First, we check for a common factor. Here there is none (other than 1 or –1).
2) Find two First terms whose product is
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
1 Factor:
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3) Find two Last terms whose product is –8.
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
1 Factor:
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4) Inspect the Outside and Inside products resulting from steps (2) and (3). Look for a combination in which the sum of the products is the middle term, –10x. 5)
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
2 Factor:
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
2
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1) First, we factor out the largest common factor, 4: 2) Now factor
3) There are four pairs of factors of 10 and each pair can be listed in two ways:
EXAMPLE Solution
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
2
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4) Look for Outside and Inside products resulting from steps (2) and (3) for which the sum is the middle term, –19x.
5)
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
In Example 2, look at the possibility Without multiplying, we can reject such a possibility. We removed the largest common factor in the first step. If 2x – 2 were one of the factors, then 2 would have to be a common factor in addition to the original 4. Thus, (2x – 2) cannot be part of the factorization of the original trinomial.
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
Tips for Factoring ax2 + bx + c, a ≠ 1
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
• Always factor out the largest common factor first, if one exists.
• Once the common factor has been factored out of the original trinomial, no binomial factor can contain a common factor (other than 1 or –1).
• If c is positive, then the signs in both binomial factors must match the sign of b. (This assumes that a > 0)
• Reversing the signs in the binomials reverses the sign of the middle term of their product.
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
Tips for Factoring ax2 + bx + c, a ≠ 1
Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
• Organize your work so that you can keep track of which possibilities have or have not been checked.
• Always check by multiplying.
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
4 Factor:
Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Other correct answers:
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
5 Factor:
Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1) Factor out a common factor, if any. There is none (other than 1 or –1). 2) Factor the first term, 6p2.
3) Factor the last term, –28q2, which has a negative coefficient.
EXAMPLE
5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the FOIL method.
5 Factor:
Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4) Try some possibilities: 5) The check is left to the students.
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Another method for factoring trinomials of the type ax2 + bx + c, a ≠ 1, involves the product, ac, of the leading coefficient a and the last term c. It is called the ac-method. Because it uses factoring by grouping, it is also referred to as the grouping method.
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
The ac-Method
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
1 Factor:
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1) First, we factor out a common factor, if any. There is none (other than 1 or –1).
2) We multiply the leading coefficient, 3, and the constant, –8:
EXAMPLE
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
1 Factor:
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3) Then we look for a factorization of in which the sum of the factors is the coefficient of the middle term, –10.
EXAMPLE
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
1 Factor:
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4) Next, we split the middle term as a sum or a difference using the factors found in step (3):
EXAMPLE
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
1 Factor:
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5) Finally, we factor by grouping, as follows:
EXAMPLE
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
1 Factor:
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
6)
EXAMPLE
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
2 Factor:
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
2
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1) First, we factor out a common factor, if any. The number 2 is common to all three terms, so we factor it out:
2) Next, we factor the trinomial . We multiply the leading coefficient and the constant, 4 and –3:
EXAMPLE Solution
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
2
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3) We try to factor –12 so that the sum of the factors is 4.
EXAMPLE Solution
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
2
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4) Then we split the middle term, 4x, as follows:
5) Finally, we factor by grouping:
EXAMPLE Solution
5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method
a Factor trinomials of the type ax2 + bx + c, a ≠ 1, using the ac-method.
2
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
6)
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.5 Factoring Trinomial Squares and Differences of Squares
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Recognize trinomial squares. b Factor trinomial squares. c Recognize differences of squares. d Factor differences of squares, being careful to factor
completely.
5.5 Factoring Trinomial Squares and Differences of Squares
a Recognize trinomial squares.
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Some trinomials are squares of binomials. For example, the trinomial x2 + 10x + 25 is the square of the binomial x + 5. A trinomial that is the square of a binomial is called a trinomial square, or a perfect square trinomial.
We can use these equations in reverse to factor trinomial squares.
5.5 Factoring Trinomial Squares and Differences of Squares
Trinomial Squares
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.5 Factoring Trinomial Squares and Differences of Squares
a Recognize trinomial squares.
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
In order for an expression to be a trinomial square: a) The two expressions A2 and B2 must be squares, such as When the coefficient is a perfect square and the
power(s) of the variable(s) is (are) even, then the expression is a perfect square.
b) There must be no minus sign before A2 or B2. c) If we multiply A and B and double the result, 2·AB, we
get either the remaining term or its opposite.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
a Recognize trinomial squares.
1 Determine whether is a trinomial square.
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.5 Factoring Trinomial Squares and Differences of Squares
Factoring Trinomial Squares
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
b Factor trinomial squares.
4 Factor:
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
b Factor trinomial squares.
6 Factor:
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.5 Factoring Trinomial Squares and Differences of Squares
b Factor trinomial squares.
6
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
b Factor trinomial squares.
7 Factor:
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
b Factor trinomial squares.
8 Factor:
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
b Factor trinomial squares.
9 Factor:
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.5 Factoring Trinomial Squares and Differences of Squares
c Recognize differences of squares.
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The following polynomials are differences of squares:
To factor a difference of squares such as x2 – 9, think about the formula we used in Chapter 4: Equations are reversible, so we also know the following.
5.5 Factoring Trinomial Squares and Differences of Squares
Difference of Squares
Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.5 Factoring Trinomial Squares and Differences of Squares
c Recognize differences of squares.
Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
In order for a binomial to be a difference of squares:
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
c Recognize differences of squares.
10 Is 9x2 – 64 a difference of squares?
Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.5 Factoring Trinomial Squares and Differences of Squares
Factoring a Difference of Squares
Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
d Factor differences of squares, being careful to factor completely.
14 Factor:
Slide 21 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
d Factor differences of squares, being careful to factor completely.
17 Factor:
Slide 22 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
d Factor differences of squares, being careful to factor completely.
19 Factor:
Slide 23 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
When no factor can be factored further, you have factored completely. Always factor completely whenever told to factor.
EXAMPLE
5.5 Factoring Trinomial Squares and Differences of Squares
d Factor differences of squares, being careful to factor completely.
21 Factor:
Slide 24 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.5 Factoring Trinomial Squares and Differences of Squares
d Factor differences of squares, being careful to factor completely.
21
Slide 25 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.5 Factoring Trinomial Squares and Differences of Squares
Tips for Factoring
Slide 26 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
• Always look first for a common factor. If there is one, factor it out.
• Be alert for trinomial squares and differences of squares. Once recognized, they can be factored without trial and error.
• Always factor completely. • Check by multiplying.
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.6 Factoring Sums or Differences of Cubes
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Factor sums or differences of cubes.
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Consider the following products:
5.6 Factoring Sums or Differences of Cubes
Sum or Difference of Cubes
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
1 Factor:
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
3 Factor:
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
3
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
4 Factor:
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
We can express this polynomial as a difference of squares.
EXAMPLE Solution
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
4
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
4
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Had we thought of factoring first as a difference of two cubes, we would have had
In this case, we might have missed some factors; can be factored as but we probably would not have known to do such factoring.
5.6 Factoring Sums or Differences of Cubes
a Factor sums or differences of cubes.
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
When you can factor as a difference of squares or a difference of cubes, factor as a difference of squares first.
5.6 Factoring Sums or Differences of Cubes
Factoring Summary
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.7 Factoring: A General Strategy
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Factor polynomials completely using any of the methods considered in this chapter.
5.7 Factoring: A General Strategy
Factoring Strategy
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To factor a polynomial: a) Always look first for a common factor. If there is one,
factor out the largest common factor. b) Then look at the number of terms. Two terms: Determine whether you have a difference
of squares, A2 – B2, or a sum or difference of cubes, A3 + B3 or A3 – B3. Do not try to factor a sum of squares: A2 + B2.
5.7 Factoring: A General Strategy
Factoring Strategy
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To factor a polynomial: b) Three terms: Determine whether the trinomial is a
square. If it is, you know how to factor. If not, try trial and error, using FOIL or the ac-method.
Four terms: Try factoring by grouping. c) Always factor completely. If a factor with more than
one term can still be factored, you should factor it. When no factor can be factored further, you have finished.
d) Check by multiplying.
EXAMPLE
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
2 Factor:
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
2
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
2
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
c) None of these factors can be factored further, so we have factored completely.
d)
EXAMPLE
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
5 Factor:
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
5
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a) We look first for a common factor: b) The expression 2 – 7xy + y2 cannot be factored. c) No further factoring is possible. d) The check is left to the students.
EXAMPLE
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
7 Factor:
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
7
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a) We look first for a common factor. There isn’t one. b) There are four terms. We try factoring by grouping:
c) Since neither factor can be factored further, we have factored completely.
d) The check is left to the students.
EXAMPLE
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
12 Factor:
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.7 Factoring: A General Strategy
a Factor polynomials completely using any of the methods considered in this chapter.
12
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a) We look first for a common factor: b) The factor 8t3 – s3 has only two terms. It is a difference
of two cubes. We factor as follows: c) No further factoring is possible. The complete
factorization is: d) The check is left to the students.
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.8 Solving Quadratic Equations by Factoring
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Solve equations (already factored) using the principle of zero products.
b Solve quadratic equations by factoring and then using the principle of zero products.
5.8 Solving Quadratic Equations by Factoring
Quadratic Equation
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
A quadratic equation is an equation equivalent to an equation of the type
5.8 Solving Quadratic Equations by Factoring
a Solve equations (already factored) using the principle of zero products.
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The product of two numbers is 0 if one or both of the numbers is 0. Furthermore, if any product is 0, then a factor must be 0. For example:
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
a Solve equations (already factored) using the principle of zero products.
1 Solve:
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
This equation will be true when either factor is 0.
Each of the numbers –3 and 2 is a solution of the original equation, as we can see in the following checks.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
a Solve equations (already factored) using the principle of zero products.
1 Solve:
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.8 Solving Quadratic Equations by Factoring
The Principle of Zero Products
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
An equation ab = 0 is true if and only if a = 0 is true or b = 0 is true, or both are true. (A product is 0 if and only if one or both of the factors is 0.)
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
a Solve equations (already factored) using the principle of zero products.
2 Solve:
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
a Solve equations (already factored) using the principle of zero products.
2
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
a Solve equations (already factored) using the principle of zero products.
2
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
4 Solve:
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
5 Solve:
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
6 Solve:
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
6
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
8 Solve:
Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
8
Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
9 Solve:
Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Remember: We must have a 0 on one side.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
9
Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
Slide 21 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The graph of y = ax2 + bx + c, a ≠ 0, is shaped like one of the following curves. Note that each x-intercept represents a solution of ax2 + bx + c = 0.
EXAMPLE
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
10 Find the x-intercepts of the graph of y = x2 – 4x – 5, shown here.
Slide 22 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
10
Slide 23 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE Solution
5.8 Solving Quadratic Equations by Factoring
b Solve quadratic equations by factoring and then using the principle of zero products.
10
Slide 24 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Thus the x-intercepts of the graph of y = x2 – 4x – 5 are (5, 0) and (–1, 0). We can now label them on the graph.
CHAPTER
5 Polynomials: Factoring
Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method 5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method 5.5 Factoring Trinomial Squares and Differences of
Squares 5.6 Factoring Sums or Differences of Cubes 5.7 Factoring: A General Strategy 5.8 Solving Quadratic Equations by Factoring
CHAPTER
5 Polynomials: Factoring
Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
5.9 Applications of Quadratic Equations
OBJECTIVES
5.9 Applications of Quadratic Equations
Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Solve applied problems involving quadratic equations that can be solved by factoring.
EXAMPLE
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
1 Kitchen Island
Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Lisa buys a kitchen island with a butcher block top as part of a remodeling project. The top of the island is a rectangle that is twice as long as it is wide and that has an area of 800 in2. What are the dimensions of the top of the island?
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
1
Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1. Familiarize. We first make a drawing. Recall that the area of a rectangle is Length·Width. We let x = the width of the top, in inches. The length is then 2x.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
1
Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
2. Translate.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
1
Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3. Solve.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
1
Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4. Check. The solutions of the equation are 20 and –20. Since the width must be positive, –20 cannot be a solution. To check 20 in., we note that if the width is 20 in., then the length is 2·20 in., or 40 in., and the area is 20 in. · 40 in., or 800 in2. Thus the solution 20 checks. 5. State. The top is 20 in. wide by 40 in. long.
EXAMPLE
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
4 Marathoner’s Numbers
Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The product of the numbers of two consecutive entrants in a marathon race is 156. Find the numbers.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
4
Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1. Familiarize. Let x = the smaller integer; then x + 1 = the larger integer.
2. Translate.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
4
Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3. Solve.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
4
Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4. Check. The solutions of the equation are 12 and –13. When x is 12, then x + 1 is 13, and 12·13 = 156. The numbers 12 and 13 are consecutive. Negative numbers are not used as entry numbers. 5. State. The entry numbers are 12 and 13.
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The problems that follow involve the Pythagorean theorem, which states a relationship involving the lengths of the sides of a right triangle. A triangle is a right triangle if it has a 90°, or right, angle. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs.
5.9 Applications of Quadratic Equations
The Pythagorean Theorem
Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then
EXAMPLE
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
5 Wood Scaffold
Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Jonah is building a wood scaffold to use for a home improvement project. He designs the scaffold with diagonal braces that are 5 ft long and that span a distance of 3 ft. How high does each brace reach vertically?
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
5
Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1. Familiarize. Let h = the height, in feet, to which each brace rises vertically.
2. Translate. A right triangle is formed, so we can use the Pythagorean theorem:
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
5
Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3. Solve.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
5
Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4. Check. Since height cannot be negative, –4 cannot be a solution. If the height is 4 ft, we have 32 + 42 = 9 + 16 = 25, which is 52. Thus, 4 checks and is the solution. 5. State. Each brace reaches a height of 4 ft.
EXAMPLE
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
6 Ladder Settings
Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
A ladder of length 13 ft is placed against a building in such a way that the distance from the top of the ladder to the ground is 7 ft more than the distance from the bottom of the ladder to the building. Find both distances.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
6
Slide 21 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1. Familiarize. The ladder and the missing distances form the hypotenuse and the legs of a right triangle. We let x = the length of the side (leg) across the bottom, in feet. Then x + 7 = the length of the other side (leg). The hypotenuse has length 13 ft.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
6
Slide 22 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
2. Translate.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
6
Slide 23 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3. Solve.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
6
Slide 24 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
3. Solve.
EXAMPLE Solution
5.9 Applications of Quadratic Equations
a Solve applied problems involving quadratic equations that can be solved by factoring.
6
Slide 25 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4. Check. The negative integer –12 cannot be the length of a side. When x = 5, x + 7 = 12, and 52 + 122 = 132. Thus, 5 and 12 check. 5. State. The distance from the top of the ladder to the ground is 12 ft. The distance from the bottom of the ladder to the building is 5 ft.
top related