· web vieweither a monomial or a sum of monomials ; each monomial is a term of the polynomial...

Chapter 8 Notes Name_______________________

Ch 8.1 – Adding and Subtracting PolynomialsMonomials: Single term that is a number, a variable, or the product of a number and a variable.Examples:

Binomials:Examples:

Trinomials:Examples:

Polynomials : Either a monomial or a sum of monomials ; Each monomial is a term of the polynomial

Standard FormWhen the polynomial is in descending order with respect to the degrees of the variables

Degree of a PolynomialThe degree of a polynomial in one variable is equal to the degree of the monomial with the greatest exponent. The degree of 3x4+5x2-7x+1 is 4.

You can name a polynomial based on its degree or the number of monomials it contains

1Mr. Keller Algebra I


Adding and Subtracting Polynomials

Group like terms by rearranging the polynomial

Examples: Add the following polynomials

5) (3x2 – 4x + 8) + (2x – 7x2 – 5) 6) (7y2 + 2y – 3) + (2 – 4y + 5y2)

Examples: Subtract the following polynomials. (REMEMBER TO DISTRIBUTE THE “NEGATIVE” TO THE QUANTITY)

7) (3x2 – 4x + 8) - (2x – 7x2 – 5) 8) (7y2 + 2y – 3) - (2 – 4y + 5y2)



Ch 8.2 – Multiplying and Factoring

Multiplying a Polynomial by a MonomialRemember the rules for exponents….. x (x2) = x 1+2 = x3

Distributive property:

x

5(x + 3) 5 (x + 3)

5(x) + 5(3) x

= __5x + 15______

Examples: Find the product of the following.1) -2x2(3x2 – 7x + 10) 2) 4a(a2 + 9)

Examples: Simplify the following expressions.3) 4(3d2 + 5d) – d(d2 -7d + 12) 4) 3(2t2 – 4t – 15) + 6t(5t + 2)



Factors and Greatest Common Factor Prime Number – number that can only be divided by itself and one; has only 2 factors

Ex: 1, 2, 3, 5, 7, 11, 13…Composite Number- number that has more than 2 factors

Ex: 4, 6, 8, 9, 10…

Prime Factorization – when a whole number is expressed as a product of prime numbersTHINK FACTOR TREE

You must factor out a -1 when factoring negative numbers.Factored Form: expressed as a product of prime numbers and variables with degree 1Example: Factor completely: 6x3y

2(3)(x)(x)(x)(y)

Greatest Common Factor – greatest term that divides into a number or monomial - factor completely and circle like factors

WHAT DOES EVERY TERM HAVE IN COMMON????? / WHAT CAN YOU DIVIDE EVERYTHING BY????Examples: Find the GCF of the following5) 18xy + 24y2 6) 10x - 15 7) -14x3y2 - 16x4y3

Examples: Factor the polynomial. (Find the GCF and divide it out)EX: 3x2 + 6x + 9 factors into 3(x2 + 2x + 3)

8) 9x + 6 9) -15x2 – 25x 10) 5k3 + 20k2 – 15k



Ch 8.3 – Multiplying Binomials (F.O.I.L.) or BOX Method

FOIL Method

FOR USE ONLY WHEN MULTIPLYING BINOMIALS

First terms in each binomial are multiplied Example: (x + 4)(x + 3)

Outer of the four terms are multiplied F O I L

Inner of the four terms are multiplied x(x) x(3) 4(x) 4(3)

Last terms of each binomial are multiplied x2 +3x +4x +12

x2 + 7x + 12

BOX MethodCreate a 2 x 2 box and label each part with a term from each binomial

(x + 4)(x + 3)x 4

x x2 + 4x + 3x + 12

3 x2 + 7x + 12

Example: Find the product of the following

1) (x + 3)(x + 5) 2) (x + 4)(x – 3) 3) (x – 5)(x + 7)


x(x) (4)x

3(x) 3(4)


4) (2n + 3g)(4n – 5g) 5) (2x – 5)(3x2 – 4x + 1)

Ch 8.4 – Multiplying Binomials - Special Cases

Square of a Sum Square of a Difference Sum and a Difference (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 (a + b)(a – b) = a2 – b2

(x + 3)2 = x2 + 2(x)(3) + (3)2

= x2 + 6x + 9What does the expression 33 mean to do? Rewrite it: Rewrite the expression x5 without exponents: How can we rewrite (x + 2)2 without exponents:

How can knowing this trick, help you with simplifying an expression then?

Example: Simplify the following6) (x + 7)2 7) (2x - 3y)2 8) (x – 4)(x + 4) 9) (3x – 9y)(3x + 9y)



8.5 – 8.8:Factoring QuadraticsFactoring By Grouping: works for polynomials with 4 terms or more; use the method above to factor

Group terms with like variables together and factor by distributive propertySteps for Factoring:

1. Factor out any common terms first (i.e….Is there anything they all divide by)2. Mutliply the First Coefficient by the Last Term (i.e. the # in front of x2 by the number with no variable)3. Find 2 factors of that product that add to the “x” term4. Rewrite the middle term as the sum of the 2 factors5. Factor by Grouping (Split the 4 terms into 2 sets of 2)6. Write the final answer as 2 quantities (x + a)(x – b)

Example: x2 + 5x + 6 1(6) = 6Factors of 6 that add to 5 = 3 and 2Rewrite: x2 + 3x + 2x + 6Grouping: x2 + 3x + 2x + 6 x(x + 3) +2(x + 3) ---Factor out the common terms on each side

(x + 3) (x + 2) ---Take out quantity from each side and write what is left

Examples: Factor the following.1) 4ax – 5a 2) 3xy + 12y

Example: Factor3) 4ab + 8b + 3a + 6 4) 3ab – 15a + 4b – 20 5) 2xy + 7x – 4y – 14

6) 2x2 + 4x + 8x + 16 7) 3x3 + 3x2 + 12x2 + 12x



WHAT HAPPENS WHEN THERE ARE NOT 4 TERMS ALREADY????Steps for Factoring:

1. Factor out any common terms first (i.e….Is there anything they all divide by)2. Create the 2x2 box again. The x2 term goes in the upper left corner and the constant term goes in the

lower right corner.3. Multiply the coefficient of the x2 and the constant term.4. Find factors of that number that add the middle term of the quadratic.5. Place those factors in the other 2 boxes with variables. 6. Factor out the GCF and complete the sides.

Example: Factor the following8) x2 + 6x + 8 9) x2 + 5x + 4 10) x2 + 7x + 12

11) x2 – 10x + 16 12) x2 – 12x + 27 13) x2 + x – 12

14) x2 – 7x – 18 15) x2 – x – 20 16) x2 – 8x - 20



Ch 8.6 – Factoring Trinomials when a ≠ 1ALWAYS LOOK FOR A GCF TO TAKE OUT FIRST!!!

Factoring Trinomials where a ≠ 1

- must be in the form ax2 + bx + c- has a coefficient in front of x2 Steps for Factoring:1. Factor out any common terms first (i.e….Is there anything they all divide by)2. Create the 2x2 box again. The x2 term goes in the upper left corner and the constant term goes in the

lower right corner.3. Multiply the coefficient of the x2 and the constant term.4. Find factors of that number that add the middle term of the quadratic.5. Place those factors in the other 2 boxes with variables. 6. Factor out the GCF and complete the sides.

Example:

7x2 + 22x + 3 → 7(3) = 21 → →

Examples:Factor

1) 5x2 + 27x + 10 2) 10x2 - 43x + 28 3) 2x2 + 7x + 5

**4) 3x2 + 24x + 45 ***5) 12x2 + 20x - 8


Factors of 21 Sum of Factors1 21 223 7 10


Ch 8.7 – Factoring Special CasesREMEMBER TO LOOK FOR GCF!!!!

Factoring a Difference of Squares Remember: (x + y)(x – y) = x2 – y2

Therefore: x2 – y2 – (x – y)(x + y

Examples: Factor the following

1) x2 – 25 2) m2 – 64 3) 4x2 – 9y2 4) 36x2 – 81y2

5) 3x3 – 27x 6) 16x2 – 16 7) 5x3 + 15x2 – 5x - 15



Perfect Square Trinomials – trinomials that are a square of a binomial first and last term are prefect squares middle term is two times the root of the first term times the root of the last term

When factoring and you multiply the 1st and last terms together:

If it is a huge number, take the square root of it, that will tell you what to fill the box with.

Examples: Factor the following.

8) 16x2 – 72x + 81 9) 4x2 + 20x + 25 10) 81r2 – 90r + 25


· web vieweither a monomial or a sum of monomials ; each monomial is a term of the polynomial...

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