Chapter 8

Polynomials and Factoring

Definitions

• Binomial- a polynomial of two terms• Degree of polynomial- the greatest value of

exponent of any term of the polynomial• Degree of monomial- the sum of the

exponents of the variables of a monomial• Difference of two squares- a difference of two

squares is an expression of the form a²-b². Factors to (a + b)(a - b)

Definitions

• Factoring by grouping- a method of factoring that uses the distributive property to remove a common binomial factor of two pairs of terms

• Monomial- a real number, a variable, or a product of a real number and one or more variables with whole-number exponents

Definitons

• Perfect square trinomial- any trinomial of the form a² + 2ab + b² or a² - 2ab + b²

• Polynomial- a monomial of the sum or difference of two or more monomials. A quotient with a variable in the denominator is not a polynomial

• Standard form of a polynomial- the form of a polynomial that places the terms in descending order by degree

• Trinomial- a polynomial of three terms

ADDING AND SUBTRACTING POLYNOMIALS

8.1

Essential Understanding

• You can use monomials to form larger expressions called polynomials. Polynomials can be added or subtracted.

Degree of monomial

• The degree of a monomial is the sum of the exponents of its variables only.

• The degree of a nonzero constant is 0.– Zero means it has no degree

Finding the Degree of Monomial

• Examples:• 11• ⅞• 12• ½• ∏• 92

Any polynomial whose largest term has an exponent that is not stated, and no variables has a degree of zero

Finding the Degree of Monomial

• Examples:• x• z• a• y• b• p

Any polynomial whose largest term has a variable with an implicit exponent of one has a degree of one

Finding the Degree of Monomial

• Examples:• Degree of 2– 2xy– 4z²– 7pq

• Degree of 3– xyz– x³– Xy²

• Degree of 4– x²y²– wxyz

Any mononomial whose largest term has an exponent or has multiple terms with exponents has the sum of the exponents as its degree

Combining Like Monomials

Example• To find degree, sum exponents• Even if bases are not the same• x²x²

– 2+2=4

• 6x³y²– 3+2=5

• -7x⁴z²– 4+2=6

• uvwxyz– 1+1+1+1+1+1=6

Explanation

• Combine exponents– Term has a degree of four

• Combine exponents– Term has a degree of five

• Combine exponents– Term has a degree of six

• Combine exponents– Term has a degree of six

Standard Form of Polynomial

Examples in Standard Form• In standard form, the

exponents (degree) descend as the terms are listed.

• 5x² + 7x - 10• 12x³ - 6x + 9• 19x⁴ - 8x³ - 5x• x⁴ + x³ + x² + x + 1

Examples NOT in Standard Form

• An expression is not in standard form if the degrees do not descend in order

• 6x-12x²• 1-3x⁴+2x+x³• 12 + 12x² + 12x⁴ + 12x

Degree of a Polynomial

• The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent.

• The degree of 3x⁴ + 5x² - 7x + 1 is four

Degree of PolynomialPolynomial Degree Name of

DegreeNumber of Terms

Name using Number of Terms

10 None (0) Constant One monomial

9237238 None (0) Constant One monomial

x One (1) Linear One monomial

7x-5 One (1) Linear Two binomial

5x² Two (2) Quadratic One monomial

12x²+144 Two (2) Quadratic Two binomial

3x²+9x+12 Two (2) Quadratic Three trinomial

8x³ Three (3) Cubic One monomial

4x³+10x Three (3) Cubic Two binomial

x³+3x²+x+1 Three (3) Cubic Four polynomial

x⁴+x³+x²+x+1 Four (4) Quartic Five polynomial

Classifying Polynomials

• To classify a polynomial1. Ensure all like terms are combined2. Find the term with the greatest exponent3. List term with greatest exponent4. Find term with next greatest exponent5. List term with next greatest exponent6. Continue process until you reach the constant

term or, if no constant term exists, the term with the least greatest exponent

MULTIPLYING AND FACTORING8.2

Essential Understanding

• You can use the distributive property to multiply a monomial by a polynomial

Multiplying Polynomials

Greatest Common Factor

MULTIPLYING BINOMIALS8.3

Essential Understanding

• There are several ways to find the product of two binomials including models, algebra, and tables.

Multiplying Binomials

Multiplying Binomials Using the Distributive Property

Multiplying Binomials Using a Table

FOIL

• FOIL stands for first outer inner last• This method does not work for multiplying

two polynomials with more than two terms for each

MULTIPLYING SPECIAL CASES8.4

Essential Understanding

• There are special rules you can use to simplify the square of a binomial or the product of a sum and difference.

• Squares of binomials have two forms:– (a + b)²– (a – b)²

The Square of a Binomial

• The square of a binomial is the square o the first term plus twice the product of the two terms plus the square of the last term.

• (a + b)² = a² + 2ab + b²• (a – b)² = a² -2ab + b²

The Product of a sum and difference

• The product of the sum and difference of the same two terms is the difference o their squares.

• (a + b)(a – b) = a² - b²

FACTORING X²+BX+C8.5

Essential Understanding

• You can write some trinomials of the form x² + bx + c as the product of two binomials

Factoring x² + bx + c

• Use a table to list the pairs of factors of the constant term c and the sums of those pairs of factors.

FACTORING AX²+BX+C8.6

Essential Understanding

• You can write some trinomials of the form ax² + bx + c as the product of two binomials

Factoring When a c is Positive∙

Factoring When a c is Negative∙

FACTORING SPECIAL CASES8.7

Essential Understanding

• You can factor some trinomials by “reversing” the rules for multiplying special case binomials that you learned in lesson 8.4

Factoring Perfect Square Trinomials

• Any trinomial of the form a² + 2ab + b² is a perfect square trinomial because it is the result of squaring a binomial.

• For example let a,b be real numbers:– (a + b)² = (a + b)(a + b) = a² + 2ab + b²– (a - b)² = (a - b)(a - b) = a² - 2ab + b²

Factoring a Difference of Two Squares

• For all real numbers a,b:– a² - b² = (a + b)(a – b)

FACTORING BY GROUPING8.8

Essential Understanding

• Some polynomials of a degree greater than 2 can be factored

Factoring Polynomials

1. Factor out the greatest common factor (GCF)2. If the polynomial has two terms or three

terms, look for a difference of two squares, a perfect square trinomial or a pair of binomial factors

3. If the polynomial has four or more terms, group terms and factor to find common binomial factors.

4. As a final check, make sure there are no common factors other than 1