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Mathematics LibreTexts

10.3: Multiply Polynomials (Part 1)

Skills to Develop

  • Multiply a polynomial by a monomial
  • Multiply a binomial by a binomial
  • Multiply a trinomial by a binomial

be prepared!

Before you get started, take this readiness quiz.

  1. Distribute: 2(x + 3). If you missed the problem, review Example 7.17.
  2. Distribute: −11(4 − 3a). If you missed the problem, review Example 7.26.
  3. Combine like terms: x2 + 9x + 7x + 63. If you missed the problem, review Example 2.21.

Multiply a Polynomial by a Monomial

In Distributive Property you learned to use the Distributive Property to simplify expressions such as 2(x − 3). You multiplied both terms in the parentheses, x and 3, by 2, to get 2x − 6. With this chapter's new vocabulary, you can say you were multiplying a binomial, x − 3, by a monomial, 2. Multiplying a binomial by a monomial is nothing new for you!

Example 10.28:

Multiply: 3(x + 7).

Solution
Distribute. CNX_BMath_Figure_10_03_001_img-01.png
  3 • x + 3 • 7
Simplify. 3x + 21

Exercise 10.55:

Multiply: 6(x + 8).

Exercise 10.56:

Multiply: 2(y + 12).

Example 10.29:

Multiply: x(x − 8).

Solution
Distribute. CNX_BMath_Figure_10_03_044_img-01.png
  x2 - 8x
Simplify. x2 - 8x

Exercise 10.57:

Multiply: y(y − 9).

Exercise 10.58:

Multiply: p(p − 13).

Example 10.30:

Multiply: 10x(4x + y).

Solution
Distribute. CNX_BMath_Figure_10_03_045_img-02.png
  10x • 4x + 10x • y
Simplify. 40x2 + 10xy

Exercise 10.59:

Multiply: 8x(x + 3y).

Exercise 10.60:

Multiply: 3r(6r + s).

Multiplying a monomial by a trinomial works in much the same way.

Example 10.31:

Multiply: −2x(5x2 + 7x − 3).

Solution
Distribute. CNX_BMath_Figure_10_03_046_img-01.png
  -2x • 5x2 + (-2x) • 7x - (-2x) • 3
Simplify. -10x3 -14x2 + 6x

Exercise 10.61:

Multiply: −4y(8y2 + 5y − 9).

Exercise 10.62:

Multiply: −6x(9x2 + x − 1).

Example 10.32:

Multiply: 4y3(y2 − 8y + 1).

Solution
Distribute. CNX_BMath_Figure_10_03_047_img-01.png
  4y3 • y2 - 4y3 • 8y + 4y3 • 1
Simplify. 4y5 -32y4 + 4y3

Exercise 10.63:

Multiply: 3x2 (4x2 − 3x + 9).

Exercise 10.64:

Multiply: 8y2 (3y2 − 2y − 4).

Now we will have the monomial as the second factor.

Example 10.33:

Multiply: (x + 3)p.

Solution
Distribute. CNX_BMath_Figure_10_03_048_img-01.png
  x • p + 3 • p
Simplify. xp + 3p

Exercise 10.65:

Multiply: (x + 8)p.

Exercise 10.66:

Multiply: (a + 4)p.

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

Using the Distributive Property

We will start by using the Distributive Property. Look again at Example 10.33.

  CNX_BMath_Figure_10_03_049_img-01.png
We distributed the p to get CNX_BMath_Figure_10_03_049_img-02.png
What if we have (x + 7) instead of p? Think of the (x + 7) as the \(\textcolor{red}{p}\) above. CNX_BMath_Figure_10_03_049_img-03.png
Distribute (x + 7). CNX_BMath_Figure_10_03_049_img-04.png
Distribute again. x2 + 7x + 3x + 21
Combine like terms. x2 + 10x + 21

Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

Be careful to distinguish between a sum and a product.

$$\begin{split} &\textbf{Sum} \qquad \qquad \qquad \quad \textbf{Product} \\ &x + x \qquad \qquad \qquad \qquad x \cdot x \\ &\; \; 2x \qquad \qquad \qquad \qquad \qquad x^{2} \\ combine\; &like\; terms \qquad add\; exponents\; of\; like\; bases \end{split}$$

Example 10.34:

Multiply: (x + 6)(x + 8).

Solution
  CNX_BMath_Figure_10_03_050_img-01.png
Distribute (x + 8). CNX_BMath_Figure_10_03_050_img-02.png
Distribute again. x2 + 8x + 6x + 48
Simplify. x2 + 14x + 48

Exercise 10.67:

Multiply: (x + 8)(x + 9).

Exercise 10.68:

Multiply: (a + 4)(a + 5).

Now we'll see how to multiply binomials where the variable has a coefficient.

Example 10.35:

Multiply: (2x + 9)(3x + 4).

Solution
Distribute (3x + 4). CNX_BMath_Figure_10_03_051_img-01.png
Distribute again. 6x2 + 8x + 27x + 36
Simplify. 6x2 + 35x + 36

Exercise 10.69:

Multiply: (5x + 9)(4x + 3).

Exercise 10.70:

Multiply: (10m + 9)(8m + 7).

In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

Example 10.36:

Multiply: (4y + 3)(6y − 5).

Solution
Distribute. CNX_BMath_Figure_10_03_052_img-01.png
Distribute again. 24y2 − 20y + 18y − 15
Simplify. 24y2 − 2y − 15

Exercise 10.71:

Multiply: (7y + 1)(8y − 3).

Exercise 10.72:

Multiply: (3x + 2)(5x − 8).

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

Example 10.37:

Multiply: (x + 2)(x − y).

Solution
Distribute. CNX_BMath_Figure_10_03_053_img-02.png
Distribute again. x2 - xy + 2x - 2y
Simplify. There are no like terms to combine.

Exercise 10.73:

Multiply: (x + 5)(x − y).

Exercise 10.74:

Multiply: (x + 2y)(x − 1).

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