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Multiplying polynomials

To multiply two polynomial a + b and c + d, think of c + d as a number, then apply the distributive law. We have:

(a + b)(c + d)
= a(c + d) + b(c + d)
= ac + ad + bc + bd

In words, to multiply two polynomial, multiply each term of one polynomial by each term of the other polynomial, and then add the products.

The procedure is illustrated by the following diagram.

Example 1
Multiply (x + 2)(x + 3)

Solution:
First multiply each term in the binomial x + 3 by the term x in the binomial x + 2, then multiply each term in the binomial x + 3 by the term 2 in the binomial x + 2, we have:
(x + 2)(x + 3)
= x² + 3x + 2x + 6
= x² + 5x + 6 combining like terms

Example 2
Multiply (2x + 3)(x² - 2x + 3)

Solution:
First multiply each term in the trinomial x² - 2x + 3 by the term 2x in the binomial 2x + 3, then multiply each term in the trinomial x² - 2x + 3 by the term 3 in the binomial 2x + 3, we have:
(2x + 3)(x² - 2x + 3)
= 2x³ - 4x² + 6x + 3x² - 6x + 9          multiplying
= 2x³ - 4x² + 3x² + 6x - 6x + 9          rearranging
= 2x³ - x² + 9           combining like terms

Example 3
Multiply (x - y)(x² + xy + y²)

Solution:
First multiply each term in the trinomial x² + xy + y² by the term x in the binomial x + y, then multiply each term in the trinomial x² + xy + y² by the term -y in the binomial x - y, we have:
(x - y)(x² + xy + y²)
= x³ + x²y + xy² - x²y - xy² - y³= x³ + x²y - x²y + xy² - xy² - y³= x³ - y³

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