Multiplying Two MonomialsTo multiply two monomials, multiply the numerical factors (the coefficients)
and then multiply the variable factors.Appendix II Polynomials A-15EXAMPLE (^1) Multiply: a. 3 y 6 y b. 3 x (^5) (2x (^5) )
StrategyWe will multiply the numerical factors and then multiply the variable
factors.
WHYThe commutative and associative properties of multiplication enable us to
reorder and regroup factors.
Solution
a. Group the numerical factors and group the variables.
Multiply: 3 6 18 and yyy^2.
b. Group the numerical factors
and group the variables.
6 x^10 Multiply: 3 2 6 and x^5 x^5 x^5 ^5 x^10.
1 3 x^5212 x^52 1 3 # 221 x^5 #x^52
18 y^2
3 y# 6 y 13 # 621 y#y 2
Self Check 1
Multiply: 7 a^3 2 a^5
Now TryProblem 15
Multiply a polynomial by a monomial.
To find the product of a polynomial and a monomial, we use the distributive property.
To multiply x4 by 3x, for example, we proceed as follows:
Use the distributive property.
Multiply the monomials: 3x(x) 3 x^2 and 3x(4) 12 x.
The results of this example suggest the following rule.
Multiplying Polynomials by Monomials
To multiply a polynomial by a monomial, multiply each term of the polynomial
by the monomial.
3 x^2 12 x
3 x 1 x 42 3 x 1 x 2 3 x 142
2
EXAMPLE (^2) Multiply: a. 2 a^2 (3a^2 4 a) b. 8 x(3x^2 2 x3)
StrategyWe will multiply each term of the polynomial by the monomial.
WHYWe use the distributive property to multiply a monomial and a polynomial.
Solution
a.
Use the distributive property.
Multiply: 2a^2 (3a^2 ) 6 a^4 and 2a^2 (4a) 8 a^3.
b.
Use the distributive property.
Multiply: 8x(3x^2 ) 24 x^3 , 8x(2x) 16 x^2 ,
and 8x(3) 24 x.
24 x^3 16 x^2 24 x
8 x(3x^2 ) 8 x(2x) 8 x(3)
8 x(3x^2 2 x3)
6 a^4 8 a^3
2 a^213 a^22 2 a^214 a 2
2 a^213 a^2 4 a 2
Self Check 2
Multiply:
a. 3 y(5y^3 4 y)
b. 5 x(3x^2 2 x3)
Now TryProblem 29