2.5. Multiplying a Monomial and a Multinomial

We will now use the things we have learned by multiplying monomials to multiply a monomial and a multinomial.

A multinomial is an algebraic expression having more than one term.

Example 55:

Some multinomials are

  1. 6x − 8

  2. 7x2 + 2x − 1

  3. x−2 + 5

Recall that a polynomial is a monomial or a multinomial in which the powers to which the variable is raised are all positive integers. The first two expressions in the preceding example are polynomials, but the third is not. The examples in this chapter will show multiplication of polynomials only, but the methods we show are valid for any multinomials. We have not yet covered the rules needed for multiplying other multinomials, such as those containing radicals, negative exponents, logarithms, and so forth. In later chapters we will show multiplication of such expressions.

A binomial is a polynomial with two terms, and a trinomial is a polynomial having three terms. In Example 55, the first expression is a binomial and the second is a trinomial.

To multiply a monomial and a multinomial, we use the distributive law (Eq. 10):

NOTE

Example 56:

These examples show the use of the distributive law.

  1. x(x + 1) = x(x) + x(1) = x2 + x

  2. 3m(m + m2) = 3m(m) + 3m(m2) = 3m2 + 3m3

  3. 2x(x + 1) = 2x(x) + 2x(1) = 2x2 + 2x

Be especially careful when multiplying negative quantities.

Example 57:

Here is an example where ...

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