§ 5.2 Multiplication of Polynomials

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.2 Multiplying PolynomialsEXAMPLE SOLUTION Multiply Rearrange factors Multiply coefficients and add exponents Simplify

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.2 Multiplying PolynomialsEXAMPLE SOLUTION Multiply Distribute Multiply coefficients and add exponents

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.2 Multiplying PolynomialsEXAMPLE SOLUTION Multiply Multiply the trinomial by each term of the binomial Multiply coefficients and add exponents Distribute Simplify Note this: We multiply each term of the binomial by each term of the trinomial. We get 6 products in all.

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.2 Multiplying Polynomials - FOIL Using the FOIL Method to Multiply Binomials first outside inside last FOIL Product of First terms Product of Outside terms Product of Inside terms Product of Last terms

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.2 Multiplying Polynomials - FOILEXAMPLE SOLUTION Multiply Combine like terms Multiply FOIL first outside inside last

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.2 Multiplying Polynomials – Special Formulas The Square of a Binomial Sum The Square of a Binomial Difference The Product of the Sum and Difference of Two Terms

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.2 Multiplying Polynomials – Special FormulasEXAMPLE SOLUTION Multiply Use the special-product formula shown. + + = Product + +

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.2 Multiplying Polynomials – Special FormulasEXAMPLE SOLUTION Multiply Use the special-product formula shown. - + = Product - +

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.2 Multiplying Polynomials – Special FormulasEXAMPLE SOLUTION Multiply Use the special-product formula shown. First Term Squared Second Term Squared Product -= =

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.2 Multiplying Polynomials – Special FormulasEXAMPLE SOLUTION Multiply We can group the terms so that the formula for the square of a binomial can be applied. (A + B) (A - B) = -

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.2 Multiplying Polynomial FunctionsEXAMPLE SOLUTION (a) (fg)(x) = f (x) g(x) (a) (fg)(x) (b) (fg)(-1) (c) (fg)(0) = (x - 4)(x + 10) FOIL Thus,

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.2 Multiplying Polynomial FunctionsCONTINUED (b) We use the product function to find (fg)(-1) – that is, the value of the function fg at -1. Replace x with -1. (c) We use the product function to find (fg)(0) – that is, the value of the function fg at 0. Replace x with 0.

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.2 Multiplying Polynomial FunctionsEXAMPLE SOLUTION To find f (a + h) – f (a), we first replace each occurrence of x with a + h and then replace each occurrence of x with a. Then we perform the resulting operations and simplify. f (a + h) – f (a) Use, replacing x with a + h and a, respectively.

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 5.2 Multiplying Polynomial Functions Combine like terms Multiply as indicated CONTINUED Multiply