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*}
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.
Example 7.13.
In each product below, we keep the same base and add the exponents.
- \(\displaystyle x^2 \cdot x^6 = x^8\)
- \(\displaystyle 5^2 \cdot 5^5 = 5^8\)
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.
- How do we simplify the product of two powers with the same base?
- How do we simplify the product of two powers with different bases?
Answer.
- We add the exponents and leave the base unchanged.
- 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
- \(\displaystyle 4^2-3b-12\)
- \(\displaystyle 4x^2+6\)
- \(\displaystyle -6x^2+27x+6\)
- \(\displaystyle 3a-8\)
- \(\displaystyle -8t^2+12t-4\)
- \(\displaystyle -30w^2+25w+30\)
Subsection Lesson
Subsubsection Activity 1: Products of Monomials
Exercises Exercises
1.
Use the first law of exponents to find each product.
- \(\displaystyle k^2 \cdot k^8\)
- \(\displaystyle y^3(y^3)\)
- \(\displaystyle a^4 \cdot a^9\)
2.
Multiply.
- \(\displaystyle 2x^2y(-3x^2y^2)\)
- \(\displaystyle -3a^4b(-4a^3b)\)
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.
- What is wrong with the statement \(5^7 \cdot 5^8 = 25^{15}\) ?
- What is the difference between the expressions \(6x^4)(-2x^4)\) and \(6x^4-2x^4\) ?
- In Activity 2 Problem 2, which of the three factors did you multiply together first?
- 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
- \(\displaystyle 3b^2-3b^3\)
- \(\displaystyle b^2-3b^3\)
- \(\displaystyle 4b^5-8b^4\)
- \(\displaystyle -18x^2y^3-12x^3y^2+30x^2y^2\)
- \(\displaystyle 6z^3-10z^2t-24zt^2+40t^3\)
- \(\displaystyle -12x^5+46x^4-14x^3\)
Exercises Homework 7.2
1.
Apply the first law of exponents to find the product.
- \(\displaystyle x^3 \cdot x^6\)
- \(\displaystyle 5^6 \cdot 5^8\)
- \(\displaystyle b^3(b)(b^5)\)
Answer.
- \(\displaystyle x^9\)
- \(\displaystyle 5^{14}\)
- \(\displaystyle b^9\)
2.
Find a value of \(n\) that makes the expressions equivalent.
- \(\displaystyle y^3 \cdot y^n=y^8\)
- \(\displaystyle a^n \cdot a^4 = a^8\)
- \(\displaystyle 3\cdot 3^n=3^3\)
3.
Simplify if possible.
- \(\displaystyle 2x^4(-3x^4)\)
- \(\displaystyle -x^4(-2x^2)\)
- \(\displaystyle -x^4 \cdot y^3\)
- \(\displaystyle -3x^5(3y^5)\)
Answer.
- \(\displaystyle -6x^8\)
- \(\displaystyle 2x^6\)
- \(\displaystyle -x^4y^3\)
- \(\displaystyle -9x^5 y^5\)
4.
Simplify if possible.
- \(\displaystyle 2x^4-3x^4\)
- \(\displaystyle -x^4-2x^2\)
- \(\displaystyle -x^4 + y^3\)
- \(\displaystyle -3x^5+3y^5\)
Exercise Group.
For the pair of expressions in Problems 5–6 find
- their product,
- their sum.
5.
\(-3z^4,~-7z^4\)
Answer.
- \(\displaystyle 21z^8\)
- \(\displaystyle -10z^4\)
6.
\(-9cd^3,~-cd^3\)
Exercise Group.
For the rectangles in Problems 7–8 find
- the perimeter,
- the area.
7.
Answer.
- \(\displaystyle 48w^5\)
- \(\displaystyle 144w^{10}\)
8.
Exercise Group.
For Problems 9–10,
- Write a product (length \(\times\) width) for the area of the rectangle.
- Use the distributive law to compute the product.
9.
Answer.
- \(\displaystyle 2a(5a+3)\)
- \(\displaystyle 10a^2+6a\)
10.
Exercise Group.
For Problems 11–14, simplify each expression if possible. If it cannot be simplified, say so.
11.
- \(\displaystyle x^2+x^2\)
- \(\displaystyle x^2(x^2)\)
- \(\displaystyle x^2-x^2\)
- \(\displaystyle x^2(-x^2)\)
Answer.
- \(\displaystyle 2x^2\)
- \(\displaystyle x^4\)
- \(\displaystyle 0\)
- \(\displaystyle -x^4\)
12.
- \(\displaystyle -x-x\)
- \(\displaystyle -x(-x)\)
- \(\displaystyle -x^2-x^2\)
- \(\displaystyle -x^2(-x^2)\)
13.
- \(\displaystyle x+x^2\)
- \(\displaystyle x(x^2)\)
- \(\displaystyle x^2-x\)
- \(\displaystyle x^2(-x)\)
Answer.
- cannot be simplified
- \(\displaystyle x^3\)
- cannot be simplified
- \(\displaystyle -x^3\)
14.
- \(\displaystyle -x^3 \cdot x\)
- \(\displaystyle x^3(-x^2)\)
- \(\displaystyle x^3-x^2\)
- \(\displaystyle (-x)^3(-x)^2\)
Exercise Group.
For Problems 15–16, multiply.
15.
\(-xy(x^2+xy+y^2)\)
Answer.
\(-x^3 y -x^2y^2 - xy^3\)
16.
\((6-st+3s^2t^2)(-3s^2t^2)\)
Exercise Group.
For Problems 17–18, simplify.
17.
\(ax(x^2+2x-3)-a(x^3+2x^2)\)
Answer.
\(-3ax\)
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.
\((x+2y)(x-y)\)
Answer.
\(x^2+xy-2y^2\)
20.
\((3s+t)(2s+3t)\)
21.
\((2x-a)(x-3a)\)
Answer.
\(2x^2-7ax+3a^2\)
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.
\(-3(x+4)(x-1)\)
Answer.
\(-3x^2-9x+12\)
24.
\(-(4x+3)(x-2)\)
25.
\(5(2x+1)(2x-1)\)
Answer.
\(20x^2-5\)
Exercise Group.
For Problems 26–29, multiply.
26.
\(4a(a-1)(a+5)\)
27.
\(s^2t^2(2s+t)(3s-t)\)
Answer.
\(6s^4t^2+s^3t^3-s^2t^4\)
28.
\((x-2)(x^2-3x+2)\)
29.
\((3x-1)(9x^2-3x+1)\)
Answer.
\(27x^3-18x^2+6x-1\)
30.
- Multiply \(~(x+1)(x+2)(x+3).\)
- Evaluate the product for \(~x=-1,~x=-2,\) and \(~x=-3)\)
31.
- Multiply \(~(x-2)(x+4)(x+5).\)
- Find three values of \(x\) for which the product is equal to zero.
Answer.
- \(\displaystyle x^3+7x^2+11x+6\)
- \(x=2\text{,}\) \(-4\text{,}\) \(-5\)
32.
Simplify mentally, without using paper, pencil, or calculator.
- \(\displaystyle 10^3(8 \cdot 10^4)\)
- \(\displaystyle (3 \cdot 10^2)(2 \cdot 20^2)\)
- \(\displaystyle (3.3 \cdot 10^2)(2 \cdot 10^2)\)
33.
The sum of two numbers is 16.
- If one of the numbers is \(n\text{,}\) write an expression for the other number.
- Write a polynomial for the product of the two numbers.
Answer.
- \(\displaystyle 16-n\)
- \(\displaystyle 16n-n^2\)
34.
A large wooden box is 3 feet longer than it is wide, and its height is 2 feet shorter than its width.
- If the width of the box is \(w\text{,}\) write expressions for its length and its height.
- Write a polynomial for the volume of the box.
- 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.
- If your first even integer is \(n\text{,}\) what is the next even integer?
- What is the next even integer after that?
- Write a polynomial for the product of the three even integers in parts (a) and (b).
Answer.
- \(\displaystyle n+2\)
- \(\displaystyle n+4\)
- \(\displaystyle n^3+6n^2+8n\)
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.
- Compute each product.
- Illustrate each product as the area of a rectangle.
36.
\((3y+4)(y+2)\)
37.
\((2w-6)(4w+3)\)
Answer.
\(8w^2-18w-18\)
38.
\((3t+5)(2t+3)\)
