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Section 7.2 Products of Polynomials

Subsection Products 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*}
Example 7.13.

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

  1. \(\displaystyle x^2 \cdot x^6 = x^8\)

  2. \(\displaystyle 5^2 \cdot 5^5 = 5^8\)

Caution 7.14.

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*}

Reading Questions Reading Questions

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?

Answer.

  1. We add the exponents and leave the base unchanged.

  2. We cannot simplify the product.

Subsection Products of Monomials

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

Example 7.15.

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*}

Reading Questions Reading Questions

2.

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

Answer.

The associative law

Subsection Multiplying by a Monomial

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

Example 7.16.

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.

Reading Questions Reading Questions

3.

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

Answer.

The distributive law

Subsection Products 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.

Example 7.17.

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 \(-1\text{.}\)

\begin{align*} (\blert{2x}\alert{-1})(3x^2-x+2) \amp = \blert{2x}(3x^2)+\blert{2x}(-x)+\blert{2x}(2)\alert{-1}(3x^2)\alert{-1}(-x)\alert{-1}(2)\\ \amp = 6x^3-2x^2+4x-3x^2+x-2 ~~~~~~ \blert{\text{Combine like terms.}}\\ \amp = 6x^3-5x^2+5x-2 \end{align*}

Reading Questions Reading Questions

4.

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

Answer.

Nine

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.

Example 7.18.

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*}

Reading Questions Reading Questions

5.

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

Answer.

The binomial factors

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

Example 7.19.

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*}

Subsection Skills Warm-Up

Exercises Exercises

Exercise Group.

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.
\(5\left[-6\left(w+\dfrac{2}{3}\right)\left(w-\dfrac{3}{2}\right)\right]\)

Subsubsection Answers to Skills Warm-Up

  1. \(\displaystyle 4^2-3b-12\)

  2. \(\displaystyle 4x^2+6\)

  3. \(\displaystyle -6x^2+27x+6\)

  4. \(\displaystyle 3a-8\)

  5. \(\displaystyle -8t^2+12t-4\)

  6. \(\displaystyle -30w^2+25w+30\)

Subsection Lesson

Subsubsection Activity 1: Products of Monomials

Exercises Exercises
1.

Use the first law of exponents to find each product.

  1. \(\displaystyle k^2 \cdot k^8\)

  2. \(\displaystyle y^3(y^3)\)

  3. \(\displaystyle a^4 \cdot a^9\)

2.

Multiply.

  1. \(\displaystyle 2x^2y(-3x^2y^2)\)

  2. \(\displaystyle -3a^4b(-4a^3b)\)

Subsubsection Activity 1: Products of Polynomials

Exercises Exercises
1.

Multiply \(~(3x+2)(3x^2+4x-2)\)

2.

Multiply \(~s^2t^2(2s+1)(3s-1)\)

3.

Multiply \(~(x+2)(3x-2)(2x-1)\)

Subsubsection Wrap-Up

Objectives.

In this Lesson we practiced the following skills:

  • Applying the first law of exponents

  • Multiplying a polynomial by a monomial

  • Computing the product of two or more polynomials

Questions.

  1. What is wrong with the statement \(5^7 \cdot 5^8 = 25^{15}\) ?

  2. What is the difference between the expressions \(6x^4)(-2x^4)\) and \(6x^4-2x^4\) ?

  3. In Activity 2 Problem 2, which of the three factors did you multiply together first?

  4. In Activity 2 Problem 3, which of the three factors did you multiply together first?

Subsection Homework Preview

Exercises Exercises

Exercise Group.

Simplify.

1.
\(2b^2(-b^2)+b(3b-b^2)\)
2.
\(2b^2-b^2(b+3b)-b^2\)
3.
\((2b-b^2)(b+3b)(-b^2)\)
Exercise Group.

Multiply.

4.
\(-6x^2y(3y^2+2xy-5y)\)
5.
\((3z-t)(2z+4t)(z-2t)\)
6.
\(-2x^3(3x-1)(2x-7)\)

Subsubsection Answers to Homework Preview

  1. \(\displaystyle 3b^2-3b^3\)

  2. \(\displaystyle b^2-3b^3\)

  3. \(\displaystyle 4b^5-8b^4\)

  4. \(\displaystyle -18x^2y^3-12x^3y^2+30x^2y^2\)

  5. \(\displaystyle 6z^3-10z^2t-24zt^2+40t^3\)

  6. \(\displaystyle -12x^5+46x^4-14x^3\)

Exercises Homework 7.2

1.

Apply the first law of exponents to find the product.

  1. \(\displaystyle x^3 \cdot x^6\)

  2. \(\displaystyle 5^6 \cdot 5^8\)

  3. \(\displaystyle b^3(b)(b^5)\)

2.

Find a value of \(n\) that makes the expressions equivalent.

  1. \(\displaystyle y^3 \cdot y^n=y^8\)

  2. \(\displaystyle a^n \cdot a^4 = a^8\)

  3. \(\displaystyle 3\cdot 3^n=3^3\)

3.

Simplify if possible.

  1. \(\displaystyle 2x^4(-3x^4)\)

  2. \(\displaystyle -x^4(-2x^2)\)

  3. \(\displaystyle -x^4 \cdot y^3\)

  4. \(\displaystyle -3x^5(3y^5)\)

4.

Simplify if possible.

  1. \(\displaystyle 2x^4-3x^4\)

  2. \(\displaystyle -x^4-2x^2\)

  3. \(\displaystyle -x^4 + y^3\)

  4. \(\displaystyle -3x^5+3y^5\)

Exercise Group.

For the pair of expressions in Problems 5–6 find

  1. their product,

  2. their sum.

5.

\(-3z^4,~-7z^4\)

6.

\(-9cd^3,~-cd^3\)

Exercise Group.

For the rectangles in Problems 7–8 find

  1. the perimeter,

  2. the area.

7.
square
8.
rectangle
Exercise Group.

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.

9.
rectangle
10.
rectangle
Exercise Group.

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

11.

  1. \(\displaystyle x^2+x^2\)

  2. \(\displaystyle x^2(x^2)\)

  3. \(\displaystyle x^2-x^2\)

  4. \(\displaystyle x^2(-x^2)\)

12.

  1. \(\displaystyle -x-x\)

  2. \(\displaystyle -x(-x)\)

  3. \(\displaystyle -x^2-x^2\)

  4. \(\displaystyle -x^2(-x^2)\)

13.

  1. \(\displaystyle x+x^2\)

  2. \(\displaystyle x(x^2)\)

  3. \(\displaystyle x^2-x\)

  4. \(\displaystyle x^2(-x)\)

14.

  1. \(\displaystyle -x^3 \cdot x\)

  2. \(\displaystyle x^3(-x^2)\)

  3. \(\displaystyle x^3-x^2\)

  4. \(\displaystyle (-x)^3(-x)^2\)

Exercise Group.

For Problems 15–16, multiply.

15.

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

16.

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

Exercise Group.

For Problems 17–18, simplify.

17.

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

18.

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

Exercise Group.

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.

\((2x-a)(x-3a)\)

Exercise Group.

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.

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

Exercise Group.

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. \(\displaystyle 10^3(8 \cdot 10^4)\)

  2. \(\displaystyle (3 \cdot 10^2)(2 \cdot 20^2)\)

  3. \(\displaystyle (3.3 \cdot 10^2)(2 \cdot 10^2)\)

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).

Exercise Group.

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.

\((3t+5)(2t+3)\)