In the third line of the solution, note that each term of has been
multiplied by each term of. This example suggests the following rule.Multiplying PolynomialsTo multiply two polynomials, multiply each term of one polynomial by each
term of the other polynomial, and then combine like terms.2 x 33 x^2 3 x 5A-18 Appendix II Polynomials
EXAMPLE (^5) Multiply:
StrategyWe will multiply each term of the trinomial, , by each term
of the binomial,.
WHYTo multiply two polynomials, we must multiply each term of one polynomial
by each term of the other polynomial.
Solution
Multiply the monomials.
Combine like terms.
Caution! The FOIL method cannot be applied here—only to products of two
binomials.
42 y^3 38 y^2 17 y 3
42 y^3 56 y^2 7 y 18 y^2 24 y 3
7 y(6y^2 ) 7 y( 8 y) 7 y(1) 3 (6y^2 ) 3 ( 8 y) 3 (1)
( 7 y3)(6y^2 8 y1)
7 y 3
6 y^2 8 y 1
(7y3)(6y^2 8 y1)
Self Check 5
Multiply:
Now TryProblem 49
(3a^2 1)(2a^4 a^2 a)
It is often convenient to multiply polynomials using a vertical formsimilar to that
used to multiply whole numbers.
Success Tip Multiplying two polynomials in vertical form is much like
multiplying two whole numbers in arithmetic.
347
25
1 735
6 940
8,675
EXAMPLE (^6) Multiply using vertical form:
a. b.
StrategyFirst, we will write one polynomial underneath the other and draw a
horizontal line beneath them. Then, we will multiply each term of the upper
polynomial by each term of the lower polynomial.
WHYVertical formmeans to use an approach similar to that used in arithmetic to
multiply two whole numbers.
(3a^2 4 a7)(2a5) (6y^3 5 y4)( 4 y^2 3)
Self Check 6
Multiply using vertical form:
a.
b.
Now TryProblem 63
( 2 x^2 3)(2x^2 4 x1)
(3x2)(2x^2 4 x5)