SECTION 5.5 Multiplying Polynomials^329
Multiply. See Section 1.8.Multiply. See Section 5.1.- 12 a 21 - 5 ab 2 106. 13 xz 214 x 2 107. 1 - m^221 m^52 108. 12 c 213 c^22
1
2
51 x+ 42 - 31 x^2 + 72 412 a+ 6 b 2 14 m- 8 n 2PREVIEW EXERCISES
OBJECTIVES OBJECTIVE 1 Multiply a monomial and a polynomial.As shown in
Section 5.1,we find the product of two monomials by using the rules for exponents
and the commutative and associative properties. Consider this example.
(^) = 72 m (^6) n 6
(^) =- 81 - 921 m (^621) n (^62)
- 8 m^61 - 9 n^62
Multiplying Polynomials
5.5
1 Multiply a
monomial and a
polynomial.
2 Multiply two
polynomials.
3 Multiply binomials
by the FOIL method.Multiplying Monomials and PolynomialsFind each product.
(a)
Distributive property
Multiply monomials.(b)
Distributive property
Multiply monomials. NOW TRYOBJECTIVE 2 Multiply two polynomials.To find the product of the polyno-
mials and we can think of as a single quantity and use the
distributive property as follows.
Distributive propertyDistributive property again
Multiply monomials.=x^3 - x^2 - 7 x- 20 Combine like terms.
=x^3 - 4 x^2 + 3 x^2 - 12 x+ 5 x- 20
=x^21 x 2 +x^21 - 42 + 3 x 1 x 2 + 3 x 1 - 42 + 51 x 2 + 51 - 42
=x^21 x- 42 + 3 x 1 x- 42 + 51 x- 42
1 x^2 + 3 x+ 521 x- 42
x^2 + 3 x+ 5 x- 4, x- 4
=- 32 m^6 - 24 m^5 - 16 m^4 + 8 m^3
+ 1 - 8 m^3212 m 2 + 1 - 8 m^321 - 12
= - 8 m^314 m^32 + 1 - 8 m^3213 m^22
- 8 m^314 m^3 + 3 m^2 + 2 m- 12
= 12 x^3 + 20 x^2
4 x^213 x+ 52 = 4 x^213 x 2 + 4 x^2152
4 x^213 x+ 52
EXAMPLE 1
CAUTION Do not confuse addition of terms with multiplication of terms.For
instance,
7 q^5 + 2 q^5 = 9 q^5 , but 17 q^5212 q^52 = 7 # 2 q^5 +^5 = 14 q^10.
NOW TRY
EXERCISE 1
Find the product.
- 3 x^512 x^3 - 5 x^2 + 102
NOW TRY ANSWER
- 6 x^8 + 15 x^7 - 30 x^5