# 9.2: Multiplication of Polynomials

**At Grade**Created by: CK-12

When multiplying polynomials together, we must remember the exponent rules we learned in the last chapter, such as the Product Rule. This rule says that if we multiply expressions that have the same base, we just add the exponents and keep the base unchanged. If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any number. We also apply the product rule on each variable separately.

Example: \begin{align*}(2x^2 y^3) \times (3x^2 y)=(2\cdot3) \times (x^2 \cdot x^2) \times (y^3 \cdot y)=6x^4 y^4\end{align*}

## Multiplying a Polynomial by a Monomial

This is the simplest of polynomial multiplications. Problems are like that of the one above.

**Example 1:** *Multiply the following monomials.*

(a) @$\begin{align*}(2x^2)(5x^3)\end{align*}@$

(c) @$\begin{align*}(3xy^5)(-6x^4y^2)\end{align*}@$

(d) @$\begin{align*}(-12a^2b^3c^4)(-3a^2b^2)\end{align*}@$

**Solution:**

(a) @$\begin{align*}(2x^2)(5x^3)=(2 \cdot 5)\cdot (x^2 \cdot x^3) = 10x^{2+3}=10x^5\end{align*}@$

(c) @$\begin{align*}(3xy^5)(-6x^4y^2)=-18x^{1+4}y^{5+2}=-18x^5y^7\end{align*}@$

(d) @$\begin{align*}(-12a^2b^3c^4)(-3a^2b^2) = 36a^{2+2}b^{3+2}c^4=36a^4b^5c^4\end{align*}@$

To multiply monomials, we use the **Distributive Property.**

**Distributive Property:** For any expressions @$\begin{align*}a, \ b\end{align*}@$, and @$\begin{align*}c\end{align*}@$, @$\begin{align*}a(b+c)=ab+ac\end{align*}@$.

This property can be used for numbers as well as variables. This property is best illustrated by an area problem. We can find the area of the big rectangle in two ways.

One way is to use the formula for the area of a rectangle.

@$$\begin{align*}Area \ of \ the \ big \ rectangle & = Length \times Width\\ Length & = a, \ Width = b + c\\ Area & = a \times (b + c)\end{align*}@$$

The area of the big rectangle can also be found by adding the areas of the two smaller rectangles.

@$$\begin{align*}Area \ of \ red \ rectangle & = ab\\ Area \ of \ blue \ rectangle & = ac\\ Area \ of \ big \ rectangle & = ab + ac\end{align*}@$$

This means that @$\begin{align*}a(b+c)=ab+ac\end{align*}@$.

In general, if we have a number or variable in front of a parenthesis, this means that each term in the parenthesis is multiplied by the expression in front of the parenthesis.

@$\begin{align*}a(b+c+d+e+f+\ldots)=ab+ac+ad+ae+af+ \ldots\end{align*}@$ The “...” means “and so on.”

**Example 2:** *Multiply @$\begin{align*}2x^3 y(-3x^4 y^2+2x^3 y-10x^2+7x+9)\end{align*}@$.*

**Solution:**

@$$\begin{align*}& 2x^3 y(-3x^4 y^2+2x^3 y-10x^2+7x+9)\\ & = (2x^3 y)(-3x^4 y^2 )+(2x^3 y)(2x^3 y)+(2x^3 y)(-10x^2 )+(2x^3 y)(7x)+(2x^3 y)(9)\\ & = -6x^7 y^3+4x^6 y^2-20x^5 y+14x^4 y+18x^3 y\end{align*}@$$

## Multiplying a Polynomial by a Binomial

A binomial is a polynomial with two terms. The Distributive Property also applies for multiplying binomials. Let’s think of the first parentheses as one term. The Distributive Property says that the term in front of the parentheses multiplies with each term inside the parentheses separately. Then, we add the results of the products.

@$\begin{align*}(a+b)(c+d)=(a+b)\cdot c+(a+b)\cdot d\end{align*}@$ Let’s rewrite this answer as @$\begin{align*}c\cdot (a+b)+d\cdot (a+b)\end{align*}@$

We see that we can apply the Distributive Property on each of the parentheses in turn.

@$$\begin{align*}c \cdot (a+b)+d\cdot (a+b)=c\cdot a+c \cdot b+d \cdot a+d \cdot b \ (\text{or} \ ca+cb+da+db)\end{align*}@$$

What you should notice is that when multiplying any two polynomials, **every term in one polynomial is multiplied by every term in the other polynomial**.

Example: *Multiply and simplify @$\begin{align*}(2x+1)(x+3)\end{align*}@$.*

Solution: We must multiply each term in the first polynomial with each term in the second polynomial. First, multiply the first term in the first parentheses by all the terms in the second parentheses.

Now we multiply the second term in the first parentheses by all terms in the second parentheses and add them to the previous terms.

Now we can simplify.

@$$\begin{align*}(2x)(x)+(2x)(3)+(1)(x)+(1)(3) & = 2x^2+6x+x+3\\ & = 2x^2+7x+3\end{align*}@$$

**Multimedia Link:** For further help, visit http://www.purplemath.com/modules/polydefs.htm – Purplemath’s website – or watch this CK-12 Basic Algebra: Adding and Subtracting Polynomials

YouTube video.

**Example 3:** *Multiply and simplify* @$\begin{align*}(4x-5)(x-20)\end{align*}@$.

**Solution:**

@$$\begin{align*}(4x)(x)+(4x)(-20)+(-5)(x)+(-5)(-20)&=4x^2-80x-5x+100\\ & = 4x^2-85x+100\end{align*}@$$

## Solving Real-World Problems Using Multiplication of Polynomials

We can use multiplication to find the area and volume of geometric shapes. Look at these examples.

**Example 4**: *Find the area of the following figure.*

Solution: We use the formula for the area of a rectangle: @$\begin{align*}\text{Area}=\text{length}\cdot\text{width}\end{align*}@$. For the big rectangle:

@$$\begin{align*}\text{Length} & = B+3, \ \text{Width}=B+2\\ \text{Area} &= (B+3)(B+2)\\ & = B^2+2B+3B+6\\ & = B^2+5B+6\end{align*}@$$

**Example 5**: *Find the volume of the following figure.*

Solution:

@$$\begin{align*}The \ volume \ of \ this \ shape & = (area \ of \ the \ base) \cdot (height).\\ \text{Area of the base} & = x(x+2)\\ & = x^2+2x\end{align*}@$$

@$$\begin{align*}Volume&=(area \ of \ base ) \times height\\ Volume&=(x^2+2x)(2x+1)\end{align*}@$$

You are asked to finish this example in the practice questions.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.

CK-12 Basic Algebra: Multiplication of Polynomials (9:49)

Multiply the following monomials.

- @$\begin{align*}(2x)(-7x)\end{align*}@$
- @$\begin{align*}4(-6a)\end{align*}@$
- @$\begin{align*}(-5a^2b)(-12a^3b^3)\end{align*}@$
- @$\begin{align*}(-5x)(5y)\end{align*}@$
- @$\begin{align*}y(xy^4)\end{align*}@$
- @$\begin{align*}(3xy^2z^2)(15x^2yz^3)\end{align*}@$

Multiply and simplify.

- @$\begin{align*}x^8 (xy^3+3x)\end{align*}@$
- @$\begin{align*}2x(4x-5)\end{align*}@$
- @$\begin{align*}6ab(-10a^2 b^3+c^5)\end{align*}@$
- @$\begin{align*}9x^3(3x^2-2x+7)\end{align*}@$
- @$\begin{align*}-3a^2b(9a^2-4b^2)\end{align*}@$
- @$\begin{align*}(x-2)(x+3)\end{align*}@$
- @$\begin{align*}(a+2)(2a)(a-3)\end{align*}@$
- @$\begin{align*}(-4xy)(2x^4 yz^3 -y^4 z^9)\end{align*}@$
- @$\begin{align*}(x-3)(x+2)\end{align*}@$
- @$\begin{align*}(a^2+2)(3a^2-4)\end{align*}@$
- @$\begin{align*}(7x-2)(9x-5)\end{align*}@$
- @$\begin{align*}(2x-1)(2x^2-x+3)\end{align*}@$
- @$\begin{align*}(3x+2)(9x^2-6x+4)\end{align*}@$
- @$\begin{align*}(a^2+2a-3)(a^2-3a+4)\end{align*}@$
- @$\begin{align*}(3m+1)(m-4)(m+5)\end{align*}@$
- Finish the volume example from Example 5 of the lesson. @$\begin{align*}Volume=(x^2+2x)(2x+1)\end{align*}@$

Find the areas of the following figures.

Find the volumes of the following figures.

**Mixed Review**

- Give an example of a fourth degree trinomial in the variable @$\begin{align*}n\end{align*}@$.
- Find the next four terms of the sequence @$\begin{align*}1,\frac{3}{2},\frac{9}{4},\frac{28}{8}, \ldots\end{align*}@$
- Reece reads three books per week.
- Make a table of values for weeks zero through six.
- Fit a model to this data.
- When will Reece have read 63 books?

- Write 0.062% as a decimal.
- Evaluate @$\begin{align*}ab\left ( a+\frac{b}{4} \right )\end{align*}@$ when @$\begin{align*}a=4\end{align*}@$ and @$\begin{align*}b=-3\end{align*}@$.
- Solve for @$\begin{align*}s\end{align*}@$: @$\begin{align*}3s(3+6s)+6(5+3s)=21s\end{align*}@$.