Multiplying Polynomials


Product of Two Monomials

To multiply two monomials together we multiply their numerical coefficients together to obtain the numerical coefficient of the answer and multiply their variable factors together using the first law of exponents. See first law of exponents

Example:

(6x² y)(2xy) = 12x³ y²

Product of a Monomial and a Polynomial

To multiply a polynomial by a monomial use the distributive property. That is multiply each of the terms of the polynomial by the monomial.

Example:

5x(2x² + 3x - 7) = 10x³ + 15x² - 35x

Product of Two Binomials

Multiplying two binomials together can be accomplished by repeatedly applying the distributive property. The four term results obtained can also be obtained more directly using what is often referred to as the FOIL METHOD. Where the letters in the word FOIL stand for first, outer, inner, and last.

                       first   outer   inner  last

(ax + b)(cx + d) = acx² + adx + bcx + bd

The terms ax and cx are referred to as the first since they are the first terms in each of the two binomials.

The terms ax and d are referred to as the outer terms.

The terms b and cx are referred to as the inner terms.

The terms b and d are referred to as the last terms since they are the last terms in each of the two binomials.

Example:

                  first  outer inner last

(5x - 7)(3x + 1)  =  15x²  +  5x  -  21x  -  7  =  15x²  -  16x  -  7

The Product of Two Polynomials Which Are Not Both Binomials

>Multiplying two polynomials together can be accomplished by repeatedly applying the distributive property. A shortcut method to obtain the same results is to multiply each of the terms of the second polynomial by each of the terms of the first polynomial.

Example:

(2x - 3)(5x² + 4x - 7) = 10x³  + 8x²  - 14x (Multiply each of the terms of the trinomial by 2x.)

                                 - 15x²  - 12x + 21 (Multiply each of the terms of the trimonial by -3.)

                                 = 10x³  - 7x²  - 26x + 21 (Combine like terms.)

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