## Section7.2Products of Polynomials

### SubsectionProducts of Powers

Suppose we would like to multiply two powers together. For instance, consider the product $(x^3)(x^4)\text{,}$ which can be written as

\begin{equation*} (x^3)(x^4)=xxx \cdot xxxx = x^7 \end{equation*}

because $x$ occurs as a factor 7 times. We see that the number of $x$'s in the product is the sum of the number of $x$'s in each factor.

On the other hand, if we'd like to multiply $x^3$ times $y^4\text{,}$ we cannot simplify the product because the two powers do not have the same base.

\begin{equation*} (x^3)(y^4)=xxx \cdot yyyy = x^3y^4 \end{equation*}

These observations illustrate the following rule.

###### First Law of Exponents.

To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,

\begin{equation*} \blert{a^m \cdot a^n = a^{m+n}} \end{equation*}
###### Example7.13.

In each product below, we keep the same base and add the exponents.

1. $x^2 \cdot x^6 = x^8$
2. $5^2 \cdot 5^5 = 5^8$
###### Caution7.14.

Each product below is $\alert{\text{incorrect}}.$

\begin{align*} 3^4 \cdot 3^3 \amp \rightarrow 9^7 \amp \amp \blert{\text{The base does not change.The correct product is}~3^7.}\\ t^3 \cdot t^5 \amp \rightarrow t^{15} \amp \amp \blert{\text{We add the exponents.The correct product is}~t^8.} \end{align*}

###### 1.
1. How do we simplify the product of two powers with the same base?
2. How do we simplify the product of two powers with different bases?

### SubsectionProducts of Monomials

We can use the first law of exponents to multiply two monomials together.

###### Example7.15.

Multiply $~~(2x^2y)(5x^4y^3)$

Solution

We rearrange the factors to group together the numerical coefficients and the powers of each base.

\begin{equation*} (2x^2y)(5x^4y^3) = (2)(5)~x^2x^4~yy^3 \end{equation*}

We multiply the coefficients together, and use the first law of exponents to find the products of the variable factors.

\begin{equation*} (2)(5)~x^2x^4~yy^3 = 10x^6y^4 \end{equation*}

###### 2.

Which law of algebra allows us to rearrange the factors in a product?

### SubsectionMultiplying by a Monomial

To multiply a polynomial by a monomial, we use the distributive law.

###### Example7.16.

Multiply $~~-3xy^2(4x^2-2xy+2)$

Solution

We apply the distributive law to multiply each term of the polynomial by the monomial $-3xy^2\text{.}$

\begin{align*} -3xy^2(4x^2-2xy+2) \amp = -3xy^2(4x^2)-3xy^2(-2xy)-3xy^2(2)\\ \amp = -3 \cdot 4 \cdot x \cdot x^2 \cdot y^2 -3(-2) \cdot x \cdot x \cdot y^2 \cdot y-3 \cdot 2 \cdot xy^2\\ \amp = -12x^3y^2+6x^2y^3-6xy^2 \end{align*}

To simplify each term, we group together the coefficients and powers with the same base.

###### 3.

Which algebraic law do we use when we multiply a polynomial by a monomial?

### SubsectionProducts of Polynomials

In Chapter 5 we found the product of two binomials using the "FOIL" method, a special case of the distributive law. We can also use the distributive law to help us compute products of two or more polynomials.

###### Example7.17.

Multiply $~~(2x-1)(3x^2-x+2)$

Solution

We multiply each term of the first polynomial by each term of the second polynomial. This involves six multiplications: We first multiply each term of the trinomial by $2x\text{,}$ then multiply each term by $-1\text{.}$

###### 4.

How many terms are there in the product of two trinomials?

###### Look Closer.

If a product contains both polynomial and monomial factors, it is a good idea to multiply the polynomial factors together first, and save the monomial factor for last.

###### Example7.18.

Multiply $~~2x(x+2)(3x-5)$

Solution

We begin by multiplying the binomial factors, $(x+2)(3x-5)\text{.}$

\begin{align*} 2x[\blert{(x+2)(3x-5)}] \amp = 2x[\blert{3x^2-5x+6x-10}]\\ \amp = 2x(3x^2+x-10) \end{align*}

Next we use the distributive law to multiply the result by $2x\text{.}$

\begin{equation*} 2x(3x^2+x-10) = 6x^3+2x^2-20x \end{equation*}

###### 5.

To compute the product $3a(a+6)(4a-1)\text{,}$ which factors should we multiply first?

In a product of three or more polynomials, we start by multiplying together any two of the three factors.

###### Example7.19.

Multiply $~~(3a-1)(a+2)(2a-3)$

Solution

We begin by multiplying together the last two binomials, $(a+2)(2a-3)\text{.}$

\begin{align*} (3a-1)[\blert{(a+2)(2a-3)}] \amp = (3a-1)[\blert{2a^2-3a+4a-6}]\\ \amp = (3a-1)(2a^2+a-6) \end{align*}

Now we use the distributive law to multiply each term of the trinomial by each term of the binomial, as shown in Example 7.17.

\begin{align*} (3a-1)(2a^2+a-6) \amp = 6a^3+3a^2-18a-2a^2-a+6~~~~ \blert{\text{Combine like terms.}}\\ \amp = 6a^3+a^2-19a+6 \end{align*}

### SubsectionSkills Warm-Up

#### ExercisesExercises

Simplify.

###### 1.
$3(b-4)-2b(3-2b)$
###### 2.
$5x-2x(1-2x)-3(x-2)$
###### 3.
$6+3\left[x-2x(x-4)\right]$
###### 4.
$a-2\left[a-2(a-2)\right]$
###### 5.
$4\left[-2\left(t-\dfrac{1}{2}\right)(t-1)\right]$
###### 6.
$5\left[-6\left(w+\dfrac{2}{3}\right)\left(w-\dfrac{3}{2}\right)\right]$

### ExercisesHomework 7.2

###### 1.

Apply the first law of exponents to find the product.

1. $x^3 \cdot x^6$
2. $5^6 \cdot 5^8$
3. $b^3(b)(b^5)$
###### 2.

Find a value of $n$ that makes the expressions equivalent.

1. $y^3 \cdot y^n=y^8$
2. $a^n \cdot a^4 = a^8$
3. $3\cdot 3^n=3^3$
###### 3.

Simplify if possible.

1. $2x^4(-3x^4)$
2. $-x^4(-2x^2)$
3. $-x^4 \cdot y^3$
4. $-3x^5(3y^5)$
###### 4.

Simplify if possible.

1. $2x^4-3x^4$
2. $-x^4-2x^2$
3. $-x^4 + y^3$
4. $-3x^5+3y^5$

For the pair of expressions in Problems 5–6 find

1. their product,
2. their sum.
###### 5.

$-3z^4,~-7z^4$

###### 6.

$-9cd^3,~-cd^3$

For the rectangles in Problems 7–8 find

1. the perimeter,
2. the area.
###### 8.

For Problems 9–10,

1. Write a product (length $\times$ width) for the area of the rectangle.
2. Use the distributive law to compute the product.
###### 10.

For Problems 11–14, simplify each expression if possible. If it cannot be simplified, say so.

###### 11.
1. $x^2+x^2$
2. $x^2(x^2)$
3. $x^2-x^2$
4. $x^2(-x^2)$
###### 12.
1. $-x-x$
2. $-x(-x)$
3. $-x^2-x^2$
4. $-x^2(-x^2)$
###### 13.
1. $x+x^2$
2. $x(x^2)$
3. $x^2-x$
4. $x^2(-x)$
###### 14.
1. $-x^3 \cdot x$
2. $x^3(-x^2)$
3. $x^3-x^2$
4. $(-x)^3(-x)^2$

For Problems 15–16, multiply.

###### 15.

$-xy(x^2+xy+y^2)$

###### 16.

$(6-st+3s^2t^2)(-3s^2t^2)$

For Problems 17–18, simplify.

###### 17.

$ax(x^2+2x-3)-a(x^3+2x^2)$

###### 18.

$3ab^2(2+3a)-2ab(3ab+2b)$

For Problems 19–21, use rectangles to help you multiply the binomials in two variables.

###### 19.

$(x+2y)(x-y)$

###### 20.

$(3s+t)(2s+3t)$

###### 21.

$(2x-a)(x-3a)$

For Problems 22–25, compute the product. Multiply the binomials together first, then multiply the result by the numerical coefficient.

###### 22.

$2(3x-1)(x-3)$

###### 23.

$-3(x+4)(x-1)$

###### 24.

$-(4x+3)(x-2)$

###### 25.

$5(2x+1)(2x-1)$

For Problems 26–29, multiply.

###### 26.

$4a(a-1)(a+5)$

###### 27.

$s^2t^2(2s+t)(3s-t)$

###### 28.

$(x-2)(x^2-3x+2)$

###### 29.

$(3x-1)(9x^2-3x+1)$

###### 30.
1. Multiply $~(x+1)(x+2)(x+3).$
2. Evaluate the product for $~x=-1,~x=-2,$ and $~x=-3)$
###### 31.
1. Multiply $~(x-2)(x+4)(x+5).$
2. Find three values of $x$ for which the product is equal to zero.
###### 32.

Simplify mentally, without using paper, pencil, or calculator.

1. $10^3(8 \cdot 10^4)$
2. $(3 \cdot 10^2)(2 \cdot 20^2)$
3. $(3.3 \cdot 10^2)(2 \cdot 10^2)$
###### 33.

The sum of two numbers is 16.

1. If one of the numbers is $n\text{,}$ write an expression for the other number.
2. Write a polynomial for the product of the two numbers.
###### 34.

A large wooden box is 3 feet longer than it is wide, and its height is 2 feet shorter than its width.

1. If the width of the box is $w\text{,}$ write expressions for its length and its height.
2. Write a polynomial for the volume of the box.
3. Write a polynomial for the surface area of the box.
###### 35.

If you count by even numbers, such as 6, 8, 10, et cetera, you are listing consecutive even integers.

1. If your first even integer is $n\text{,}$ what is the next even integer?
2. What is the next even integer after that?
3. Write a polynomial for the product of the three even integers in parts (a) and (b).

In Lesson 7.3 we'll use rectangles to factor quadratic trinomials. Problems 36-38 review representing a product as the area of a rectangle.

1. Compute each product.
2. Illustrate each product as the area of a rectangle.
###### 36.

$(3y+4)(y+2)$

###### 37.

$(2w-6)(4w+3)$

###### 38.

$(3t+5)(2t+3)$