Section 6.8 Exponents and Polynomials Chapter Review
Subsection 6.8.1 Introduction to Exponent Rules
In Section 6.1 we covered the rules of exponents for multiplication.
Example 6.8.1.
Simplify the following expressions using the rules of exponents:
\(\displaystyle -2t^3\cdot 4t^5\)
\(\displaystyle 5\left(v^4\right)^2\)
\(\displaystyle -(3u)^2\)
\(\displaystyle (-3z)^2\)
\(\displaystyle -2t^3\cdot 4t^5=-8t^8\)
\(\displaystyle 5\left(v^4\right)^2=5v^8\)
\(\displaystyle -(3u)^2=-9u^2\)
\(\displaystyle (-3z)^2=9z^2\)
Subsection 6.8.2 Exponent Rules
In Section 6.2 we covered the exponent rules and how to use them.
Example 6.8.2. Quotients and Exponents.
Let \(t\) and \(q\) be real numbers, where \(q \neq 0\) and \(t \neq 0\text{.}\) Find another way to write \(\left(\frac{q^9}{t\cdot q^3}\right)^2\text{.}\)
We first use the quotient rule, then the quotient to a power rule, then the power to a power rule.
Example 6.8.3. The Zero Exponent.
Recall that the zero exponent rule says that any real number raised to the \(0\)-power is \(1\text{.}\) Using this, and the other exponent rules, find another way to write \(-9^0\text{.}\)
Remember that in expressions like \(-9^0\text{,}\) the exponent only applies to what it is directly next to! In this case, the \(0\) only applies to the \(9\) and not the negative sign. So,
Example 6.8.4. Negative Exponents.
Write \(5x^{-3}\) without any negative exponents.
Recall that the negative exponent rule says that a factor in the numerator with a negative exponent can be flipped into the denominator. So
Note that the \(5\) does not move to the denominator because the \(-3\) exponent only applies to the \(x\) to which it is directly attached.
Example 6.8.5. Summary of Exponent Rules.
Use the exponent rules in List 6.2.14 to write the expressions in a different way. Reduce and simplify when possible. Always find a way to write your final simplification without any negative exponents.
\(\displaystyle \dfrac{24p^3}{20p^{12}}\)
\(\displaystyle \left(\dfrac{2v^5}{4g^{-2}}\right)^4\)
\(\displaystyle 12n^7\left(m^0\cdot n^2\right)^2\)
\(\displaystyle \dfrac{k^5}{k^{-4}}\)
\(\displaystyle \begin{aligned}[t] \frac{24p^3}{20p^{12}}\amp=\frac{24}{20}\cdot\frac{p^3}{p^{12}}\\ \amp=\frac{6}{5}\cdot p^{3-12}\\ \amp=\frac{6}{5}\cdot p^{-9}\\ \amp=\frac{6}{5}\cdot\frac{1}{p^9}\\ \amp=\frac{6}{5p^9}\end{aligned} \)
\(\displaystyle \begin{aligned}[t] \left(\frac{2v^5}{4g^{-2}}\right)^4\amp=\left(\frac{v^5}{2g^{-2}}\right)^4\\ \amp=\left(\frac{v^5g^2}{2}\right)^4\\ \amp=\frac{v^{5\cdot4}g^{2\cdot4}}{2^4}\\ \amp=\frac{v^{20}g^{8}}{16}\end{aligned} \)
\(\displaystyle \begin{aligned}[t] \amp12n^7\left(m^0\cdot n^2\right)^2\\ \amp=12n^7\left(1\cdot n^2\right)^2\\ \amp=12n^7\left(n^2\right)^2\\ \amp=12n^7n^{2\cdot 2}\\ \amp=12n^7n^4\\ \amp=12n^{7+4}\\ \amp=12n^{11}\end{aligned} \)
\(\displaystyle \begin{aligned}[t] \frac{k^5}{k^{-4}}\amp=k^5\cdot k^{4}\\ \amp=k^{5+4}\\ \amp=k^9\end{aligned} \)
Subsection 6.8.3 Scientific Notation
In Section 6.3 we covered the definition of scientific notation, how to convert to and from scientific notation, and how to do some calculations in scientific notation.
Example 6.8.6. Scientific Notation for Large Numbers.
The distance to the star Betelgeuse is about \(3{,}780{,}000{,}000{,}000{,}000\) miles. Write this number in scientific notation.
The gross domestic product (GDP) of California in the year 2017 was about \(\$2.746\times10^{12}\text{.}\) Write this number in standard notation.
\(3{,}780{,}000{,}000{,}000{,}000=3.78\times10^{15}\text{.}\)
\(\$2.746\times10^{12}=\$2{,}746{,}000{,}000{,}000\text{.}\)
Example 6.8.7. Scientific Notation for Small Numbers.
Human DNA forms a double helix with diameter \(2\times10^{-9}\) meters. Write this number in standard notation.
A single grain of Forget-me-not (Myosotis) pollen is about \(0.00024\) inches in diameter. Write this number in scientific notation.
\(2\times10^{-9}=0.000000002\text{.}\)
\(0.00024=2.4\times10^{-4}\text{.}\)
Example 6.8.8. Multiplying and Dividing Using Scientific Notation.
The fastest spacecraft so far have traveled about \(5\times10^6\) miles per day.
If that spacecraft traveled at that same speed for \(2\times10^4\) days (which is about \(55\) years), how far would it have gone? Write your answer in scientific notation.
The nearest star to Earth, besides the Sun, is Proxima Centauri, about \(2.5\times10^{13}\) miles from Earth. How many days would you have to fly in that spacecraft at top speed to reach Proxima Centauri
-
Remember that you can find the distance traveled by multiplying the rate of travel times the time traveled: \(d=r\cdot t\text{.}\) So this problem turns into
\begin{align*} d\amp=r\cdot t\\ d\amp=\left(\highlight{5}\times10^6\right)\cdot \left(\highlight{2}\times10^4\right)\\ \end{align*}Multiply coefficient with coefficient and power of \(10\) with power of \(10\text{.}\)
\begin{align*} \amp=\left(\highlight{5}\cdot\highlight{2}\right) \left(10^6\times10^4\right)\\ \amp=\highlight{10}\times10^{10}\\ \end{align*}Remember that this still isn't in scientific notation. So we convert like this:
\begin{align*} \amp=\highlight{1.0\times10^1}\times10^{10}\\ \amp=1.0\times10^{11} \end{align*}So, after traveling for \(2\times10^4\) days (55 years), we will have traveled about \(1.0\times10^{11}\) miles. That's one-hundred million miles. I hope someone remembered the snacks.
-
Since we are looking for time, let's solve the equation \(d=r\cdot t\) for \(t\) by dividing by \(r\) on both sides: \(t=\frac{d}{r}\text{.}\) So we have:
\begin{align*} t\amp=\frac{d}{r}\\ t\amp=\frac{2.5\times10^{13}}{5\times10^6}\\ \end{align*}Now we can divide coefficient by coefficient and power of \(10\) with power of \(10\text{.}\)
\begin{align*} t\amp=\frac{2.5}{5}\times\frac{10^{13}}{10^6}\\ t\amp=\highlight{0.5}\times10^7\\ t\amp=\highlight{5\times10^{-1}}\times10^7\\ t\amp=5\times10^6 \end{align*}This means that to get to Proxima Centauri, even in our fastest spacecraft, would take \(5\times10^6\) years. Converting to standard form, this is \(5{,}000{,}000\) years. I think we're going to need a faster ship.
Subsection 6.8.4 Adding and Subtracting Polynomials
In Section 6.4 we covered the definitions of a polynomial, a term of a polynomial, a coefficient of a term, the degree of a term, the degree of a polynomial, the leading term of a polynomial, a constant term, monomials, binomials, and trinomials, and how to write a polynomial in standard form.
Example 6.8.9. Polynomial Vocabulary.
Decide if the following statements are true or false.
The expression \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is a polynomial.
The expression \(4x^6-3x^{-2}-x+1\) is a polynomial.
The degree of the polynomial \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is \(10\text{.}\)
The degree of the term \(5x^2y^4\) is \(6\text{.}\)
The leading coefficient of \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is \(\frac{3}{5}\text{.}\)
There are 4 terms in the polynomial \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\text{.}\)
The polynomial \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is in standard form.
True. The expression \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is a polynomial.
False. The expression \(4x^6-3x^{-2}-x+1\) is not a polynomial. Variables are only allowed to have whole number exponents in polynomials and the second term has a \(-2\) exponent.
False. The degree of the polynomial \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is not \(10\text{.}\) It is \(7\text{,}\) which is the highest power of any variable in the expression.
True. The degree of the term \(5x^2y^4\) is \(6\text{.}\)
False. The leading coefficient of \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is not \(\frac{3}{5}\text{.}\)The leading coefficient comes from the degree \(7\) term which is \(-\frac{1}{5}\text{.}\)
True. There are 4 terms in the polynomial \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\text{.}\)
False. The polynomial \(\frac{3}{5}x^2-\frac{1}{5}x^7+\frac{x}{2}-4\) is not in standard form. The exponents have to be written from highest to lowest, i.e. \(-\frac{1}{5}x^7+\frac{3}{5}x^2+\frac{x}{2}-4\text{.}\)
Example 6.8.10. Adding and Subtracting Polynomials.
Simplify the expression \(\left(\frac{2}{9}x-4x^2-5\right)+\left(6x^2-\frac{1}{6}x-3\right)\text{.}\)
First, identify like terms and group them either physically or mentally. Then we will look for common denominators for these like terms and combine appropriately.
Subsection 6.8.5 Multiplying Polynomials
In Section 6.5 we covered how to multiply two polynomials together using distribution, FOIL, and generic rectangles.
Example 6.8.11. Multiplying Binomials.
Expand the expression \((5x-6)(3+2x)\) using the binomial multiplication method of your choice: distribution, FOIL, or generic rectangles.
We will show work using the FOIL method.
Example 6.8.12. Multiplying Polynomials Larger than Binomials.
Expand the expression \((3x-2)\left(4x^2-2x+5\right)\) by multiplying every term in the first factor with every term in the second factor.
\(\begin{aligned}[t]\amp(\highlight{3x}\mathbin{\lighthigh{-}}\mathbin{\lighthigh{2}})\left(4x^2-2x+5\right)\\ \amp=\highlight{3x}\cdot4x^2+\highlight{3x}\cdot(-2x)+\highlight{3x}\cdot5+(\mathbin{\lighthigh{-}}\mathbin{\lighthigh{2}})\cdot4x^2+(\mathbin{\lighthigh{-}}\mathbin{\lighthigh{2}})\cdot(-2x)+(\mathbin{\lighthigh{-}}\mathbin{\lighthigh{2}})\cdot5\\ \amp=12x^3-6x^2+15x-8x^2+4x-10\\ \amp=12x^3-14x^2+19x-10\end{aligned} \)
Subsection 6.8.6 Special Cases of Multiplying Polynomials
In Section 6.6 we covered how to square a binomial and how to find the product of the sum or difference of two terms.
Example 6.8.13. Squaring a Binomial.
Recall that Fact 6.6.3 gives formulas that help square a binomial.
Simplify the expression \((2x+3)^2\text{.}\)
Remember that you can use FOIL to do these problems, but in the interest of understanding concepts at a higher level for use in later chapters, we will use the relevant formula from Fact 6.6.3. In this case, since we have a sum of two terms being squared, we will use \((a+b)^2 = a^2+2ab+b^2\text{.}\)
First, identify \(\highlight{a}\) and \(\lighthigh{b}\text{.}\) In this case, \(\highlight{a=2x}\) and \(\lighthigh{b=3}\text{.}\) So, we have:
Example 6.8.14. The Product of the Sum and Difference of Two Terms.
Recall that Fact 6.6.11 gives a formula to help multiply things that look like \((a+b)(a-b)\text{.}\)
Simplify the expression \((7x+4)(7x-4)\text{.}\)
Remember that you can use FOIL to do these problems, but in the interest of understanding concepts at a higher level for use in later chapters, we will use the formula from Fact 6.6.11. In this case, that means we will use \((a+b)(a-b) = a^2-b^2\text{.}\)
First, identify \(\highlight{a}\) and \(\lighthigh{b}\text{.}\) In this case, \(\highlight{a=7x}\) and \(\lighthigh{b=4}\text{.}\) So, we have:
Example 6.8.15. Binomials Raised to Other Powers.
To raise binomials to powers higher than \(2\text{,}\) we start by expanding the expression and multiplying all factors together from left to right.
Expand the expression \((2x-5)^3\text{.}\)
Subsection 6.8.7 Dividing by a Monomial
In Section 6.7 we covered how you can split a fraction up into multiple terms if there is a sum or difference in the numerator. Mathematically, this happens using the rule \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\text{.}\) This formula can be used for any number of terms in the numerator, and for both sums and differences.
Example 6.8.16.
Expand the expression \(\frac{12x^5+2x^3-4x^2}{4x^2}\text{.}\)
Exercises 6.8.8 Exercises
1.
Use the properties of exponents to simplify the expression.
\({t^{20}}\cdot{t^{11}}\)
2.
Use the properties of exponents to simplify the expression.
\({y^{4}}\cdot{y^{4}}\)
3.
Use the properties of exponents to simplify the expression.
\(\displaystyle{\left(r^{4}\right)^{10}}\)
4.
Use the properties of exponents to simplify the expression.
\(\displaystyle{\left(x^{5}\right)^{7}}\)
5.
Use the properties of exponents to simplify the expression.
\(\displaystyle{\left(2r\right)^3}\)
6.
Use the properties of exponents to simplify the expression.
\(\displaystyle{\left(5t\right)^3}\)
7.
Use the properties of exponents to simplify the expression.
\(\displaystyle{({-4y^{14}})\cdot({-3y^{9}})}\)
8.
Use the properties of exponents to simplify the expression.
\(\displaystyle{({9x^{16}})\cdot({4x^{3}})}\)
9.
Use the properties of exponents to simplify the expression.
\(\displaystyle{{\left(-5p^{4}\right)^{2}}=}\)
\(\displaystyle{{-\left(5p^{4}\right)^{2}}=}\)
10.
Use the properties of exponents to simplify the expression.
\(\displaystyle{{\left(-7q^{3}\right)^{2}}=}\)
\(\displaystyle{{-\left(7q^{3}\right)^{2}}=}\)
11.
Simplify the following expression.
\({\left(2x^{2}\right)^{2}\!\left(x^{2}\right)^{2}}\)=
12.
Simplify the following expression.
\({\left(4x^{4}\right)^{3}\!\left(x^{5}\right)^{2}}\)=
13.
Simplify the following expressions if possible.
\(\displaystyle{ {-2a+3a}=}\)
\(\displaystyle{ (-2a)(3a)=}\)
\(\displaystyle{ {-2a-3a^{3}}=}\)
\(\displaystyle{ (-2a)(-3a^{3})=}\)
14.
Simplify the following expressions if possible.
\(\displaystyle{ {4c-2c}=}\)
\(\displaystyle{ (4c)(-2c)=}\)
\(\displaystyle{ {4c-2c^{3}}=}\)
\(\displaystyle{ (4c)(-2c^{3})=}\)
15.
Multiply the polynomials.
\({7r^{2}}\left({-10r^{2}+3r}\right)=\)
16.
Multiply the polynomials.
\({-4r^{2}}\left({-8r^{2}+8r}\right)=\)
Exponent Rules
17.
Use the properties of exponents to simplify the expression.
\(\left(4y^{10}\right)^4\)
18.
Use the properties of exponents to simplify the expression.
\(\left(5x^{11}\right)^3\)
19.
Use the properties of exponents to simplify the expression.
\(\displaystyle{({5x^{20}})\cdot({-2x^{6}})}\)
20.
Use the properties of exponents to simplify the expression.
\(\displaystyle{({9y^{3}})\cdot({3y^{18}})}\)
21.
Use the properties of exponents to simplify the expression.
\(\displaystyle{\left({-\frac{y^{6}}{3}}\right) \cdot \left({-\frac{y^{12}}{2}}\right)}\)
22.
Use the properties of exponents to simplify the expression.
\(\displaystyle{\left({-\frac{y^{8}}{6}}\right) \cdot \left({\frac{y^{5}}{3}}\right)}\)
23.
Use the properties of exponents to simplify the expression.
\(\left(-29\right)^0=\)
24.
Use the properties of exponents to simplify the expression.
\(\left(-24\right)^0=\)
25.
Use the properties of exponents to simplify the expression.
\(-34^0=\)
26.
Use the properties of exponents to simplify the expression.
\(-39^0=\)
27.
Use the properties of exponents to simplify the expression.
\(\left(\displaystyle\frac{-9}{8x^{7}}\right)^{2}=\)
28.
Use the properties of exponents to simplify the expression.
\(\left(\displaystyle\frac{-9}{2x^{2}}\right)^{2}=\)
29.
Use the properties of exponents to simplify the expression.
\(\displaystyle\frac{{3t^{12}}}{{15t^{2}}}=\)
30.
Use the properties of exponents to simplify the expression.
\(\displaystyle\frac{{4t^{19}}}{{16t^{15}}}=\)
31.
Use the properties of exponents to simplify the expression.
\(\left(\displaystyle\frac{x^{2}}{2y^{5}z^{6}}\right)^{3}=\)
32.
Use the properties of exponents to simplify the expression.
\(\left(\displaystyle\frac{x^{8}}{2y^{4}z^{10}}\right)^{3}=\)
33.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{ \left(\frac{1}{6}\right)^{-2} }\)
34.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{ \left(\frac{1}{7}\right)^{-2} }\)
35.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{\displaystyle {15x^{-10}}= }\)
36.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{\displaystyle {10x^{-11}}= }\)
37.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{\displaystyle {\frac{20}{x^{-12}}}= }\)
38.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{\displaystyle {\frac{15}{x^{-3}}}= }\)
39.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{\displaystyle {\frac{16x^{-6}}{x^{-9}}}= }\)
40.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle{\displaystyle {\frac{6x^{-8}}{x^{-27}}}= }\)
41.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle\frac{y^{-5}}{\left(y^{8}\right)^{8}}=\)
42.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle\frac{r^{-4}}{\left(r^{5}\right)^{6}}=\)
43.
Rewrite the expression simplified and using only positive exponents.
\(r^{-19}\cdot r^{8}=\)
44.
Rewrite the expression simplified and using only positive exponents.
\(t^{-12}\cdot t^{9}=\)
45.
Rewrite the expression simplified and using only positive exponents.
\((7t^{-6})\cdot (3t^{2})=\)
46.
Rewrite the expression simplified and using only positive exponents.
\((5t^{-18})\cdot (7t^{11})=\)
47.
Rewrite the expression simplified and using only positive exponents.
\(\left(-5x^{-12}\right)^{-2}\)
48.
Rewrite the expression simplified and using only positive exponents.
\(\left(-2x^{-5}\right)^{-2}\)
49.
Rewrite the expression simplified and using only positive exponents.
\(\left(2y^{13}\right)^{4}\cdot y^{-29}=\)
50.
Rewrite the expression simplified and using only positive exponents.
\(\left(4y^{9}\right)^{2}\cdot y^{-6}=\)
51.
Rewrite the expression simplified and using only positive exponents.
\(\left(r^{11}x^{3}\right)^{-4}=\)
52.
Rewrite the expression simplified and using only positive exponents.
\(\left(r^{4}t^{12}\right)^{-4}=\)
53.
Rewrite the expression simplified and using only positive exponents.
\(\left(t^{-9}y^{8}\right)^{-4}=\)
54.
Rewrite the expression simplified and using only positive exponents.
\(\left(t^{-15}x^{5}\right)^{-4}=\)
55.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle\left(\frac{t^{12}}{4}\right)^{-3}=\)
56.
Rewrite the expression simplified and using only positive exponents.
\(\displaystyle\left(\frac{x^{7}}{3}\right)^{-4}=\)
Scientific Notation
57.
Write the following number in scientific notation.
\(270=\)
58.
Write the following number in scientific notation.
\(37000=\)
59.
Write the following number in scientific notation.
\(0.0048=\)
60.
Write the following number in scientific notation.
\(0.058=\)
61.
Write the following number in decimal notation without using exponents.
\(6.78\times 10^{4}=\)
62.
Write the following number in decimal notation without using exponents.
\(7.77\times 10^{3}=\)
63.
Write the following number in decimal notation without using exponents.
\(8.77\times 10^{0}=\)
64.
Write the following number in decimal notation without using exponents.
\(9.77\times 10^{0}=\)
65.
Write the following number in decimal notation without using exponents.
\(1.77\times 10^{-4}=\)
66.
Write the following number in decimal notation without using exponents.
\(2.76\times 10^{-2}=\)
67.
Multiply the following numbers, writing your answer in scientific notation.
\((4\times 10^{3})(6\times 10^{4})=\)
68.
Multiply the following numbers, writing your answer in scientific notation.
\((5\times 10^{5})(3\times 10^{3})=\)
69.
Divide the following numbers, writing your answer in scientific notation.
\(\displaystyle\frac{4.8\times 10^{2}}{6\times 10^{-5}}=\)
70.
Divide the following numbers, writing your answer in scientific notation.
\(\displaystyle\frac{4.2\times 10^{4}}{7\times 10^{-3}}=\)
Adding and Subtracting Polynomials
71.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-14r^{10}-19r^{6}}}\) is a
monomial
binomial
trinomial
72.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{13t^{4}-2t^{2}}}\) is a
monomial
binomial
trinomial
73.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-t^{10}+17t^{9}+15t^{8}}}\) is a
monomial
binomial
trinomial
74.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-16x^{10}+5x^{9}-9x^{4}}}\) is a
monomial
binomial
trinomial
75.
Find the degree of the following polynomial.
\(\displaystyle{ {-13x^{6}y^{5}+11xy^{4}+8x^{2}-6} }\)
76.
Find the degree of the following polynomial.
\(\displaystyle{ {-8x^{6}y^{7}-3x^{2}y^{2}-16x^{2}-18} }\)
77.
Add the polynomials.
\(\displaystyle{\left({-5x^{2}+6x-3}\right)+\left({-2x^{2}+8x+6}\right)}\)
78.
Add the polynomials.
\(\displaystyle{\left({6x^{2}+10x-3}\right)+\left({-6x^{2}+3x-6}\right)}\)
79.
Add the polynomials.
\(\displaystyle{\left({-5r^{6}-4r^{4}-10r^{2}}\right)+\left({6r^{6}+3r^{4}-10r^{2}}\right)}\)
80.
Add the polynomials.
\(\displaystyle{\left({2r^{6}-10r^{4}+3r^{2}}\right)+\left({-10r^{6}+3r^{4}+2r^{2}}\right)}\)
81.
Add the polynomials.
\(\displaystyle{\left({9x^{3}+3x^{2}-4x+{\frac{7}{4}}}\right)+\left({9x^{3}+4x^{2}+2x+{\frac{1}{6}}}\right)}\)
82.
Add the polynomials.
\(\displaystyle{\left({-10x^{3}-7x^{2}+4x+{\frac{9}{8}}}\right)+\left({-2x^{3}+10x^{2}-8x+{\frac{1}{4}}}\right)}\)
83.
Subtract the polynomials.
\(\displaystyle{\left({-9x^{2}+10x}\right)-\left({-10x^{2}-8x}\right)}\)
84.
Subtract the polynomials.
\(\displaystyle{\left({-7x^{2}+3x}\right)-\left({-x^{2}+7x}\right)}\)
85.
Subtract the polynomials.
\(\displaystyle{\left({-4x^{2}+9x-5}\right)-\left({-2x^{2}+5x-1}\right)}\)
86.
Subtract the polynomials.
\(\displaystyle{\left({5x^{2}-4x-6}\right)-\left({-6x^{2}-5x+7}\right)}\)
87.
Subtract the polynomials.
\(\displaystyle{\left({7r^{6}-2r^{4}+10r^{2}}\right)-\left({4r^{6}-7r^{4}-9r^{2}}\right)}\)
88.
Subtract the polynomials.
\(\displaystyle{\left({-4r^{6}-8r^{4}-3r^{2}}\right)-\left({4r^{6}+8r^{4}-2r^{2}}\right)}\)
89.
Add or subtract the given polynomials as indicated.
\(\left({8x^{3}-6xy-4y^{9}}\right)-\left({3x^{3}+5xy-5y^{9}}\right)\)
90.
Add or subtract the given polynomials as indicated.
\(\left({9x^{2}-10xy+6y^{4}}\right)-\left({-3x^{2}-6xy-10y^{4}}\right)\)
91.
A farmhand is building two pig pens of the same width, but different lengths. If \(x\) is the common width, in meters, then the cost of building one pen is \({26x^{2}+2.5x-33.5}\) dollars, and the cost of building the other pen is \({49x^{2}-2.5x+42}\) dollars. Whatâs the total cost of building those two pens?
A polynomial representing the total cost of building those two pens is dollars.
92.
A farmhand is building two pig pens of the same width, but different lengths. If \(x\) is the common width, in meters, then the cost of building one pen is \({46x^{2}-6x-34.5}\) dollars, and the cost of building the other pen is \({22x^{2}+6x+14}\) dollars. Whatâs the total cost of building those two pens?
A polynomial representing the total cost of building those two pens is dollars.
Multiplying Polynomials
93.
Multiply the polynomials.
\({-7x}\left({x-6}\right)=\)
94.
Multiply the polynomials.
\({-4x}\left({x+4}\right)=\)
95.
Multiply the polynomials.
\({-9y^{2}}\left({-9y^{2}+10y-2}\right)=\)
96.
Multiply the polynomials.
\({-6r^{2}}\left({-6r^{2}+5r-3}\right)=\)
97.
Multiply the polynomials.
\(\left({3r+9}\right)\left({r+3}\right)=\)
98.
Multiply the polynomials.
\(\left({8r+3}\right)\left({r+10}\right)=\)
99.
Multiply the polynomials.
\(\left({t+5}\right)\left({t-3}\right)=\)
100.
Multiply the polynomials.
\(\left({t+1}\right)\left({t-9}\right)=\)
101.
Multiply the polynomials.
\(\left({5x-9}\right)\left({x-6}\right)=\)
102.
Multiply the polynomials.
\(\left({3x-1}\right)\left({6x-10}\right)=\)
103.
Multiply the polynomials.
\({3\!\left(x+2\right)\!\left(x+3\right)}=\)
104.
Multiply the polynomials.
\({-3\!\left(x+2\right)\!\left(x+3\right)}=\)
105.
Multiply the polynomials.
\({x\!\left(x-2\right)\!\left(x+2\right)}=\)
106.
Multiply the polynomials.
\({-x\!\left(x+2\right)\!\left(x+3\right)}=\)
107.
Multiply the polynomials.
\(\displaystyle{({a+8b})({a^{2}-2ab+10b^{2}}) = }\)
108.
Multiply the polynomials.
\(\displaystyle{({a-9b})({a^{2}-6ab-10b^{2}}) = }\)
109.
A rectangleâs length is \(10\) feet shorter than \(5 \text{ times}\) its width. If we use \(w\) to represent the rectangleâs width, use a polynomial to represent the rectangleâs area in expanded form.
\(\displaystyle{ \text{area}=}\) square feet
110.
A rectangleâs length is \(1\) feet shorter than \(4 \text{ times}\) its width. If we use \(w\) to represent the rectangleâs width, use a polynomial to represent the rectangleâs area in expanded form.
\(\displaystyle{ \text{area}=}\) square feet
Special Cases of Multiplying Polynomials
111.
Expand the square of a binomial.
\(\left({4x+8}\right)^2=\)
112.
Expand the square of a binomial.
\(\left({10y+2}\right)^2=\)
113.
Expand the square of a binomial.
\(\left({y-5}\right)^2=\)
114.
Expand the square of a binomial.
\(\left({r-8}\right)^2=\)
115.
Expand the square of a binomial.
\(\displaystyle{ ({7a-9b})^2 = }\)
116.
Expand the square of a binomial.
\(\displaystyle{ ({8a+6b})^2 = }\)
117.
Multiply the polynomials.
\(\left({t-9}\right)\left({t+9}\right)=\)
118.
Multiply the polynomials.
\(\left({t+9}\right)\left({t-9}\right)=\)
119.
Multiply the polynomials.
\(\left({5-2x}\right)\left({5+2x}\right)=\)
120.
Multiply the polynomials.
\(\left({2+6x}\right)\left({2-6x}\right)=\)
121.
Multiply the polynomials.
\(\left({5y^{7}+10}\right)\left({5y^{7}-10}\right)=\)
122.
Multiply the polynomials.
\(\left({4y^{5}-6}\right)\left({4y^{5}+6}\right)=\)
123.
Simplify the given expression into an expanded polynomial.
\(\left({r+2}\right)^3=\)
124.
Simplify the given expression into an expanded polynomial.
\(\left({r+5}\right)^3=\)
Dividing by a Monomial
Simplify the following expressions
125.
\(\frac{{-21r^{12}+42r^{10}}}{{7}}\)
126.
\(\frac{{-60t^{20}-40t^{15}}}{{5}}\)
127.
\(\frac{{10t^{19}-6t^{17}+24t^{14}}}{{2t^{3}}}\)
128.
\(\frac{{24x^{19}-48x^{11}-24x^{10}}}{{-6x^{3}}}\)
129.
\(\frac{{117x^{8}-27x^{6}}}{{9x}}\)
130.
\(\frac{{20y^{15}-55y^{11}}}{{5y}}\)