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Section 5.7 Exponents and Polynomials Chapter Review

Adding and Subtracting Polynomials.

A polynomial is a mathematical expression involving one or more variables where terms are added together. Each term must either be a number or be in the form of a number (possibly \(1\)) multiplied by the variable(s) raised to whole number powers.
The coefficient of a term in a polynomial is the numerical factor in that term.
A term in a polynomial with no variable factor is called a constant term.
When a term only has one variable, its degree is the exponent on that variable. When a term has more than one variable, its degree is the sum of the exponents on the variables. A constant term has degree \(0\text{.}\) For a polynomial overall, its degree is the largest degree among the terms of that polynomial.
The leading term of a polynomial is the term with the greatest degree (assuming there is no tie). The coefficient of a polynomial’s leading term is called the polynomial’s leading coefficient.
The standard form of a polynomial is when the polynomial is written with the terms appearing in descending order.
There are special words for polynomials that have one term (monmial), two terms (binmial), or three terms (trinmial). There are special words for describing a polynomial of degree \(0\) (constant), degree \(1\) (linear), degree \(2\) (quadratic), and degree \(3\) (cubic)
Polynomials can be added or subtracted together by combining any like terms from the two polynomials. For example, with \(\left(4x^2+\frac{2}{9}x-5\right)+\left(6x^2-\frac{1}{6}x-3\right)\text{,}\) we can add the quadratic terms \(4x^2\) and \(6x^2\) to get \(10x^2\text{.}\) We can add the linear terms \(\frac{2}{9}x\) and \(-\frac{1}{6}x\) to get \(\frac{1}{18}x\text{.}\) And we can add the constant terms \(-5\) and \(-3\) to get \(-8\text{.}\) So the result is \(10x^2+\frac{1}{18}x-8\text{.}\)
Be careful when subtracting polynomials to “distribute” the subtraction over the terms from the second polynomial. For example
\begin{align*} \amp\left(4x^2+\frac{2}{9}x-5\right)-\left(6x^2-\frac{1}{6}x-3\right)\\ \amp=4x^2+\frac{2}{9}x-5-6x^2+\frac{1}{6}x+3\\ \amp=-2x^2+\frac{7}{18}x-2 \end{align*}
Notice all the signs that changed in the second step.

Checkpoint 5.7.1.

What specific type of polynomial is the given polynomial? What is its degree? What is its leading term? What is its leading coefficient?
\({-48.9x^{4}-91.5-61.5x}\)
Explanation.
This is a trinomial because it has \(3\) terms. Its degree is \({4}\) because that is the highest exponent on \(x\) in the terms. The leading term is \({-48.9x^{4}}\) because that’s the term with degree \({4}\text{.}\) The leading coefficent is \({-48.9}\) because that’s the coefficient of the leading term.

Checkpoint 5.7.2.

Add the polynomials.
\(\left({-7y^{7}-9y^{5}}\right)+\left({3y^{7}-y^{5}}\right)\)
Explanation.
\begin{equation*} \begin{aligned} \amp\left({-7y^{7}-9y^{5}}\right)+\left({3y^{7}-y^{5}}\right)\\ \amp=\left({-7y^{7}+3y^{7}}\right)+\left({-9y^{5}-y^{5}}\right)\\ \amp={-4y^{7}-10y^{5}} \end{aligned} \end{equation*}

Checkpoint 5.7.3.

Subtract the polynomials.
\(\left({-7.6y^{4}+y-7.2y^{8}}\right)-\left({8.7y^{8}+0.6y^{4}-4.7y^{9}}\right)\)
Explanation.
\begin{equation*} \begin{aligned} \amp\left({-7.6y^{4}+y-7.2y^{8}}\right)-\left({8.7y^{8}+0.6y^{4}-4.7y^{9}}\right)\\ \amp=\left({-7.6y^{4}+0.6y^{4}}\right)+{y}+\left({-7.2y^{8}+8.7y^{8}}\right)+{-4.7y^{9}}\\ \amp={-8.2y^{4}+y-15.9y^{8}+4.7y^{9}} \end{aligned} \end{equation*}

Introduction to Exponent Properties.

In an expression like \(2^3\text{,}\) we call \(2\) the base and \(3\) the exponent. The whole thing (which works out to equal \(8\)) is called a power.
There are certain properties of exponent expressions that can be used to simplify or otherwise rewrite those expressions. Let \(x,\) and \(y\) represent real numbers, variables, or algebraic expressions, and let \(m\) and \(n\) represent positive integers. Then the following properties hold:
Product of Powers
\(\displaystyle x^m\cdot x^n=x^{m+n}\)
Power to Power
\(\displaystyle (x^m)^n=x^{m\cdot n}\)
Product to Power
\(\displaystyle (xy)^n = x^n\cdot y^n\)

Checkpoint 5.7.4.

Use the properties of exponents to simplify the expression.
\({p^{10}p^{5}}\)
Explanation.
We have two powers with the same base. So we can keep that base, and add the exponents: \({p^{15}}\text{.}\)

Checkpoint 5.7.5.

Use the properties of exponents to simplify the expression.
\({\left(t^{9}\right)^{7}}\)
Explanation.
This expression raises \(t\) to an exponent, and then raises the result to another exponent. In this situation we can multiply the two exponents and get: \({t^{63}}\text{.}\)

Checkpoint 5.7.6.

Use the properties of exponents to simplify the expression.
\({\left(3v^{2}\right)^{2}}\)
Explanation.
This expression is a product of two things (\(3\) and \(v^{2}\)) that is then raised to a power. We can raise each of the two factors to that outer power instead:
\begin{equation*} 3^{2}\left(v^{2}\right)^{2} \end{equation*}
and then use the power-to-power property of exponents to simplify it more: \({9v^{4}}\text{.}\)

Dividing by a Monomial.

You can split a fraction up into multiple terms if there is a sum or difference in the numerator and a monomial in the denominator. Symbolically:
\begin{equation*} \frac{a+b}{c}=\frac{a}{c}+\frac{b}{c} \end{equation*}
This works with subtraction in the numerator as well. And it naturally extends to when there are more than two terms.
In this section, we also added one more exponent property:
Quotient of Powers Property
\(\dfrac{a^m}{a^n} = a^{m-n}\) (when \(m\gt n\))
That is, when you divide a power by a power, and they use the same base, then the result is a power using the same base, with the difference of the exponents.

Checkpoint 5.7.7.

Use properties of exponents to simplify the expression.
\(\displaystyle{\frac{63y^{9}}{9y^{6}}}\)
Explanation.
We can divide the coefficients to get \(7\text{.}\) With the powers of \(y\text{,}\) we can subtract the exponents to get \(y^3\text{.}\) The result is \({7y^{3}}\text{.}\)

Checkpoint 5.7.8.

Simplify the expression.
\(\displaystyle{\frac{-9k^{12}+4k^{9}-7k^{11}}{k^{3}}}\)
Explanation.
We can divide \(k^{3}\) into each of the three terms to get \({-9k^{9}+4k^{6}-7k^{8}}\text{.}\)

Checkpoint 5.7.9.

A rectangular prism’s base area can be calculated by the formula \(B=\frac{V}{h}\text{,}\) where \(V\) is the volume and \(h\) is the height.
A certain rectangular prism’s volume can be modeled by \({8x^{6}-12x^{3}-20x}\) cubic units. If its height is \({2x}\) units, find the prism’s base area.
Explanation.
Using the formula,
\begin{equation*} B=\frac{{8x^{6}-12x^{3}-20x}}{{2x}} \end{equation*}
Now we can divide \({2x}\) into each of the three terms in the numerator. So the base area is modeled by \({4x^{5}-6x^{2}-10}\text{.}\)

Multiplying Polynomials.

To multiply a monomial by a polynomial, you distribute the monomial to each term in the polynomial. For example:
To multiply a binomial by a binomial, you can use the “FOIL” acronym for “First”, “Outer”, “Inner”, “Last”. For each letter, pair up appropriate terms from the two monomials and multiply them together to get one term in the result. For example:
We can extend this to multiplying polynomials with more than two terms. Each term from the first polynomial will multiply with each term from the second.

Checkpoint 5.7.10.

Multiply the monomial with the binomial, writing the result as a single simplified polynomial.
\({-5x}\left({x+10}\right)\)
Explanation.
We multiply the monomial by each term in the binomial, using the properties of exponents to help us.
\(\begin{aligned} {-5x}({x+10})\amp ={-5x^{2}-50x} \end{aligned}\)

Checkpoint 5.7.11.

Multiply the binomial with the binomial, writing the result as a single simplified polynomial.
\(\left({y+6}\right)\left({y+4}\right)\)
Explanation.
We use the “FOIL” technique: First Outside Inside Last.
\(\begin{aligned} \left({y+6}\right)({y+4})\amp ={y^{2}+4y+6y+24}\\ \amp ={y^{2}+10y+24} \end{aligned}\)

Checkpoint 5.7.12.

Multiply the polynomials together, writing the result as a single simplified polynomial.
\(\left({-3x-5}\right)\left({5x^{3}+3x^{2}+4x-5}\right)\)
Explanation.
We multiply the first term in the binomial by each term in the polynomial, and then multiply the second term in the binomial by each term in the polynomial; then we combine like terms.
\(\begin{aligned} \left({-3x-5}\right)({5x^{3}+3x^{2}+4x-5})\amp ={-15x^{4}-9x^{3}-12x^{2}+15x-25x^{3}-15x^{2}-20x+25}\\ \amp ={-15x^{4}-34x^{3}-27x^{2}-5x+25} \end{aligned}\)

Special Cases of Multiplying Polynomials.

There are a few situations where you need to multiply two polynomials together and if you recognize certain patterns, you can almost immediately write the result instead of taking more time (and having more opportunities for human error).
When two binomials are multiplied together where one is the sum of two terms and the other is the difference of the same two terms, the result is just the first term squared minus the second term squared.
\begin{equation*} (a+b)(a-b)=a^2-b^2 \end{equation*}
When a binomial is squared, the result can be written by taking these steps.
  • Square the first term.
  • Square the second term.
  • In between, multiply the two terms together and double that.
If a binomial is raised to a higher exponent than \(2\text{,}\) it may help to break the power into pieces that are raised to the second power and then multiply the results. For example when the exponent is \(3\text{:}\)
\begin{equation*} (a+b)^3=(a+b)(a+b)^2 \end{equation*}
We could square \((a+b)\) and get \((a+b)\left(a^2+2ab+b^2\right)\) and then multiply this trinomial and binomial together, no longer trying to use any special pattern recognition.

Checkpoint 5.7.13.

Square the binomial, writing the result as a single expanded polynomial.
\(\left({r-2}\right)^2\)
Explanation.
We observe that this is the square of a binomial, so we use the pattern:
\(\displaystyle{(a-b)^2=a^2-2ab+b^2}\)
and write
\(\begin{aligned} \left({r-2}\right)({r-2})\amp =r^2-2\cdot2\cdot r+2^2\\ \amp ={r^{2}-4r+4} \end{aligned}\)

Checkpoint 5.7.14.

Multiply the polynomials, writing the result as a single expanded polynomial.
\(\left({4r+2}\right)\left({4r-2}\right)\)
Explanation.
We observe that this is the product of the sum of two terms with the difference of the same two terms.
\begin{equation*} \begin{aligned} \left({4r+2}\right)({4r-2})\amp =(4r)^2-2^2\\ \amp ={16r^{2}-4} \end{aligned} \end{equation*}

Checkpoint 5.7.15.

Simplify the given expression into an expanded polynomial.
\(\left({t+2}\right)^3\)
Explanation.
We can write \(\left({t+2}\right)^3\) as
\(\left({t+2}\right)^3 = \left({t+2}\right)\left({t+2}\right)^2\)
This means that we can use the squared binomial pattern, and then multiply the first factor \({t+2}\) by the result
\(\begin{aligned} \left({t+2}\right)^3 \amp = \left({t+2}\right)\left({t+2}\right)^2\\ \amp =\left({t+2}\right)\left({t^{2}+4t+4}\right)\\ \amp = {t^{3}+4t^{2}+4t+2t^{2}+8t+8}\\ \amp ={t^{3}+6t^{2}+12t+8} \end{aligned}\)

More Exponent Properties.

In addition to the four exponent properties already listed in this review, there are four more to be familar with. If \(a\) and \(b\) are real numbers, and \(m\) and \(n\) are integers, then we have the following properties:
Quotient to a Power Property
\(\left( \dfrac{a}{b} \right)^{m} = \dfrac{a^{m}}{b^{m}}\text{,}\) as long as \(b \neq 0\)
Zero Exponent Definition
\(\displaystyle a^{0} = 1\)
Negative Exponent Definition
\(\displaystyle a^{-m} = \frac{1}{a^m}\)
Negative Exponent Reciprocal Property
\(\displaystyle \frac{1}{a^{-m}} = a^m\)
Use the properties of exponents to simplify the expression. If there are any exponents in your answer, they should be positive.

Checkpoint 5.7.16.

\(\left(\displaystyle\frac{x^{9}}{10}\right)^{2}\)
Explanation.
The quotient to power rule says that
\(\left(\displaystyle\frac{a}{b}\right)^m = \displaystyle\frac{a^m}{b^m}\)
Also remember that
\((a^m)^n = a^{mn}\)
Using these two tools together allows us to write
\(\begin{aligned} \left(\displaystyle\frac{x^{9}}{10}\right)^{2} \amp = \displaystyle\frac{x^{9\cdot2}}{10^{2}}\\ \amp = \displaystyle\frac{x^{18}}{100}\\ \end{aligned}\)

Checkpoint 5.7.17.

\(-3^0\)
Explanation.
The zero-exponent definition says that \(a^0=1\) where \(a\) is any real number; notice that the exponent applies to \(3\text{,}\) not to \(\left(-3\right)\) so
\(\begin{aligned} -3^0 \amp = (-1)\cdot \left(3\right)^0\\ \amp = -1\cdot 1\\ \amp = -1 \end{aligned}\)

Checkpoint 5.7.18.

\(\displaystyle {\frac{7}{x^{-4}}}\)
Explanation.
The factor with the negative exponent in the denominator can be moved to the numerator, but now with a positive exponent: \({7x^{4}}\text{.}\)

Exercises Review Exercises for Chapter 5

Section 1: Adding and Subtracting Polynomials

Exercise Group.
What specific type of polynomial is the given polynomial? What is its degree? What is its leading term? What is its leading coefficient?
1.
\({-5z^{6}+6z^{3}}\)
2.
\({{\frac{3}{7}}z^{3}t^{9} - {\frac{1}{6}}z^{8}t^{3} - 4z^{7}t^{7}}\)
Exercise Group.
Add the polynomials.
3.
\(\left({7y^{6}+4y-8}\right)+\left({-6y^{6}-y-2}\right)\)
4.
\(\left({{\frac{5}{8}}r+{\frac{2}{5}}r^{7} - {\frac{1}{5}}r^{8}}\right)+\left({{\frac{7}{4}}r-r^{8}}\right)\)
Exercise Group.
Subtract the polynomials.
5.
\(\left({-2r-3}\right)-\left({-5r+6}\right)\)
6.
\(\left({-{\frac{9}{5}}t - {\frac{9}{4}}}\right)-\left({-{\frac{1}{2}}t - {\frac{3}{4}}}\right)\)
7.
\(\left({4t^{7}-9t^{2}}\right)-\left({t^{7}-7t^{8}}\right)\)
Exercise Group.
Add or subtract the multivariable polynomials as indicated.
8.
\(\left({2x^{6}z^{7}-x^{4}z^{5}}\right)-\left({8x^{6}z^{7}+9x^{4}z^{5}}\right)\)
9.
\(\left({{\frac{3}{2}}x^{2}y^{3} - {\frac{4}{9}}xy^{2}+{\frac{3}{7}}y^{2}}\right)+\left({{\frac{5}{2}}x^{2}y^{3} - {\frac{2}{5}}xy^{2} - {\frac{2}{7}}y^{5}}\right)\)
10.
\(\left({-9.5y^{5}r^{7}+5.2y^{9}r^{3}-0.5y^{6}}\right)-\left({-6.6y^{6}+3.6y^{5}r^{7}-2y^{4}r^{9}}\right)\)
Exercise Group.
Evaluate the given polynomial at the given numbers.
11.
\({2t^{2}+4t-2}\)
for \(t=-4\text{,}\) \(-5\text{,}\) and \(5\)
12.
\({5y^{2}+y^{7}}\)
for \(y=-2\text{,}\) \(-1\text{,}\) and \(2\)
13.
A company that sells a certain product can use polynomials to model its sales revenue (in dollars) and its costs (also in dollars). The variable \(x\) represents the number of units produced. Profit is revenue minus costs.
If revenue is modeled by \({4.1x^{2}+4.5x+4166}\) and costs are modeled by \({3.9x^{2}+1.45x+1191}\text{,}\) what is the model for profit?

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Exercise Group.
Use the properties of exponents to simplify the expression.
14.
\({7\cdot 7^{4}}\)
15.
\({v^{10}v^{6}v^{5}}\)
16.
\({\left(4x\right)^{4}}\)
17.
\({\left(3af\right)^{4}}\)
18.
\({8d^{7}\cdot 7d^{5}}\)
19.
\({-\left(4f^{5}\right)^{2}}\)
Exercise Group.
Use the properties of exponents together with the distributive property to write the expression in a new simplified way where there are no grouping symbols.
20.
\({-7i\mathopen{}\left(4i-8\right)}\)
21.
\({2k^{8}\mathopen{}\left(10k^{2}+6k^{9}\right)}\)
Simplifying Expressions.
Simplify the given expression.
22.
\({4p^{3}\mathopen{}\left(3p^{2}\right)^{4}}\)
23.
\({4r^{8}c^{3}\mathopen{}\left(3r^{5}c\right)^{4}}\)
24.
\({2v\cdot 2v^{8}+9v^{3}\cdot 9v^{6}}\)
25.
\({3x\mathopen{}\left(7x^{5}\right)^{3}+\left(8x^{2}\right)^{2}\cdot 4x^{12}}\)

Section 3: Dividing by a Monomial

Quotient of Powers Property.
Use properties of exponents to simplify the expression.
26.
\(\displaystyle{\frac{a^{3}}{a^{2}}}\)
27.
\(\displaystyle{\frac{-18c^{7}}{-6c^{5}}}\)
28.
\(\displaystyle{\frac{124^{536}}{124^{296}}}\)
29.
\(\displaystyle{\frac{8i^{30}r^{11}}{20i^{20}r^{2}}}\)
Dividing Polynomials by Monomials.
Simplify the expression.
30.
\(\displaystyle{\frac{35i^{11}+40i^{12}}{5i^{6}}}\)
31.
\(\displaystyle{\frac{-24j^{18}-36j^{15}-48j^{16}}{6j^{9}}}\)
32.
\(\displaystyle{\frac{14v^{6}-49v^{7}}{7v}}\)
33.
\(\displaystyle{\frac{56n^{16}i^{12}+16n^{14}i^{9}}{8n^{7}i^{6}}}\)
34.
A cylinder’s height can be calculated by the formula \(h=\frac{V}{B}\text{,}\) where \(V\) is the volume and \(B\) is the base area.
A certain cylinder’s volume can be modeled by \({20\pi x^{7}-30\pi x^{4}-25\pi x^{3}}\) cubic units. If its base area is \({5\pi x^{2}}\) square units, find the cylinder’s height.

Section 4: Multiplying Polynomials

Multiplying Monomials with Binomials.
Multiply the monomial with the binomial, writing the result as a single simplified polynomial.
35.
\({2x}\left({9x+5}\right)\)
36.
\({-7x^{2}}\left({-8x^{2}-6x}\right)\)
37.
\({-4x^{6}y^{7}}\left({-8x^{15}+4y^{4}}\right)\)
38.
\({5a^{8}}\left({-3a^{4}-4a^{6}b^{7}+2b^{6}}\right)\)
Multiplying Binomials.
Multiply the binomial with the binomial, writing the result as a single simplified polynomial.
39.
\(\left({y+6}\right)\left({y-5}\right)\)
40.
\(\left({3r-1}\right)\left({4r-8}\right)\)
41.
\(\left({6r-7}\right)\left({2r^{2}-8}\right)\)
42.
\(\left({a+9b}\right)\left({a-9b}\right)\)
43.
\(\left({10ab+3}\right)\left({3ab+4}\right)\)
44.
\({x\mathopen{}\left(x+7\right)\mathopen{}\left(x+3\right)}\)
Multiplying Larger Polynomials.
Multiply the polynomials together, writing the result as a single simplified polynomial.
45.
\(\left({-2x-2}\right)\left({x^{2}+2x+5}\right)\)
46.
\(\left({a+b-4}\right)\left({a+b+4}\right)\)
47.
A rectangle’s length is \(4\) feet shorter than \(5 \text{ times}\) its width. If we use \(w\) to represent the rectangle’s width, use a polynomial to represent the rectangle’s area in expanded form.
48.
A rectangle’s base can be modeled by \({x-6}\) meters, and its height can be modeled by \({x-4}\) meters. Use a polynomial to represent the rectangle’s area in expanded form.

Section 5: Special Cases of Multiplying Polynomials

Perfect Square Trinomial Formula.
Square the binomial, writing the result as a single expanded polynomial.
49.
\(\left({r+1}\right)^2\)
50.
\(\left({8r-9}\right)^2\)
51.
\(\left({9a+5b}\right)^2\)
Difference of Squares Formula.
Multiply the polynomials, writing the result as a single expanded polynomial.
52.
\(\left({t-12}\right)\left({t+12}\right)\)
53.
\(\left({x^{5}+6}\right)\left({x^{5}-6}\right)\)
54.
\(({3x+5y})({3x-5y})\)
55.
\(\left({2x-6}\right)\left({2x+6}\right)\)
56.
\(({x^{2}+5y^{2}})({x^{2}-5y^{2}})\)
57.
\(({6x^{8}y^{8}-4y^{2}})({6x^{8}y^{8}+4y^{2}})\)
Binomials Raised to Other Powers.
Simplify the given expression into an expanded polynomial.
58.
\(\left({r-6}\right)^3\)
59.
\(\left({5r-2}\right)^3\)

Section 6: More Exponent Rules

Simplifying Expressions with Exponents.
Use the properties of exponents to simplify the expression. If there are any exponents in your answer, they should be positive.
60.
\(\left(3x^{10}\right)^2\)
61.
\(-2\left(-4x^{11}\right)^3\)
62.
\(2^0+\left(-2\right)^0\)
63.
\(\left(-779t\right)^0\)
64.
\(\left(\displaystyle\frac{x^{9}}{2y^{10}z^{5}}\right)^{3}\)
65.
\(\displaystyle\frac{4^{-3}}{5^{-2}}\)
66.
\(\displaystyle {\frac{7x^{-7}}{x}}\)
67.
\(\displaystyle {\frac{8x^{-18}}{9x^{-33}}}\)
68.
\(\displaystyle\frac{1}{26r^{-13}}\)
69.
\(\displaystyle\frac{11t^{2}}{15t^{9}}\)
70.
\(t^{-17}\cdot t^{8}\)
71.
\(\left(-10\right)^{-2}\)
72.
\(2^{-2}\)
73.
\(-3^{-3}\)
74.
\(\displaystyle\left(\frac{y^{10}}{y^{3}}\right)^{-2}\)
75.
\(\left(5y^{-4}\right)^{-3}\)
76.
\(\left(3r^{12}\right)^{4}\cdot r^{-31}\)
77.
\(\left(r^{9}\right)^{-3}\)
78.
\(\left(t^{-13}x^{4}\right)^{-5}\)
79.
\(\displaystyle\frac{\left(t^{4}r^{-7}\right)^{-4}}{\left(t^{-8}r^{8}\right)^{-2}}\)
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