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Identify Polynomials, Monomials, Binomials, and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
ⓐ
$81{b}^{5}-24{b}^{3}+1$
ⓑ
$5{c}^{3}+11{c}^{2}-c-8$
ⓒ
$\frac{14}{15}y+\frac{1}{7}$
ⓓ 5
ⓔ
$4y+17$
ⓐ trinomial ⓑ polynomial ⓒ binomial ⓓ monomial ⓔ binomial
ⓐ
${x}^{2}-{y}^{2}$
ⓑ
$\mathrm{-13}{c}^{4}$
ⓒ
${x}^{2}+5x-7$
ⓓ
${x}^{2}{y}^{2}-2xy+8$
ⓔ 19
ⓐ
$8-3x$
ⓑ
${z}^{2}-5z-6$
ⓒ
${y}^{3}-8{y}^{2}+2y-16$
ⓓ
$81{b}^{5}-24{b}^{3}+1$
ⓔ
$\mathrm{-18}$
ⓐ binomial ⓑ trinomial ⓒ polynomial ⓓ trinomial ⓔ monomial
ⓐ
$11{y}^{2}$
ⓑ
$\mathrm{-73}$
ⓒ
$6{x}^{2}-3xy+4x-2y+{y}^{2}$
ⓓ
$4y+17$
ⓔ
$5{c}^{3}+11{c}^{2}-c-8$
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
ⓐ
$6{a}^{2}+12a+14$
ⓑ
$18x{y}^{2}z$
ⓒ
$5x+2$
ⓓ
${y}^{3}-8{y}^{2}+2y-16$
ⓔ
$\mathrm{-24}$
ⓐ 2 ⓑ 4 ⓒ 1 ⓓ 3 ⓔ 0
ⓐ
$9{y}^{3}-10{y}^{2}+2y-6$
ⓑ
$\mathrm{-12}{p}^{4}$
ⓒ
${a}^{2}+9a+18$
ⓓ
$20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30$
ⓔ 17
ⓐ
$14-29x$
ⓑ
${z}^{2}-5z-6$
ⓒ
${y}^{3}-8{y}^{2}+2y-16$
ⓓ
$23a{b}^{2}-14$
ⓔ
$\mathrm{-3}$
ⓐ 1 ⓑ 2 ⓒ 3 ⓓ 3 ⓔ 0
ⓐ
$62{y}^{2}$
ⓑ 15
ⓒ
$6{x}^{2}-3xy+4x-2y+{y}^{2}$
ⓓ
$10-9x$
ⓔ
${m}^{4}+4{m}^{3}+6{m}^{2}+4m+1$
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
${\phantom{\rule{0.2em}{0ex}}\text{7x}}^{2}+5{x}^{2}$
$12{x}^{2}$
${\phantom{\rule{0.2em}{0ex}}\text{4y}}^{3}+6{y}^{3}$
$\mathrm{-3}m+9m$
$\text{\u2212}y-5y$
$13z-(\mathrm{-4}z)$
$\mathrm{-10}x-35x$
$\text{14x}-3y-13x$
$5{u}^{2}+4{v}^{2}-6{u}^{2}$
$p{q}^{2}-4p-3{q}^{2}$
${x}^{2}y-3x+7x{y}^{2}$
$\text{19y}+5z$
Add: $\phantom{\rule{0.2em}{0ex}}\text{4x},3y,\mathrm{-3}x$
Subtract $5{x}^{6}\text{from}-12{x}^{6}$ .
$\mathrm{-17}{x}^{6}$
Subtract $2{p}^{4}\text{from}-7{p}^{4}$ .
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
$(4{y}^{2}+10y+3)+(8{y}^{2}-6y+5)$
$({x}^{2}+6x+8)+(\mathrm{-4}{x}^{2}+11x-9)$
$\mathrm{-3}{x}^{2}+17x-1$
$({y}^{2}+9y+4)+(\mathrm{-2}{y}^{2}-5y-1)$
$(7{x}^{2}-9x+2)+(6{x}^{2}-4)$
$({p}^{2}-6p-18)+(2{p}^{2}+11)$
$(3{b}^{2}-4b+1)-(5{b}^{2}-b-2)$
$({b}^{2}-7b+5)-({b}^{2}-2b+9)$
$(10{r}^{2}-20r)-(r-8)$
Subtract $\left(9{x}^{2}+2\right)$ from $\left(12{x}^{2}-x+6\right)$ .
$3{x}^{2}-x+4$
Subtract $\left(5{y}^{2}-y+12\right)$ from $\left(10{y}^{2}-8y-20\right)$ .
Subtract $\left(7{w}^{2}-4w+2\right)$ from $\left(8{w}^{2}-w+6\right)$ .
${w}^{2}+3w+4$
Subtract $\left(5{x}^{2}-x+12\right)$ from $\left(9{x}^{2}-6x-20\right)$ .
Find the sum of $\left(2{p}^{3}-8\right)$ and $\left({p}^{2}+9p+18\right)$ .
$2{p}^{3}+{p}^{2}+9p+10$
Find the sum of
$\left({q}^{2}+4q+13\right)$ and
$\left(7{q}^{3}-3\right)$ .
Find the sum of $\left(8{a}^{3}-8a\right)$ and $\left({a}^{2}+6a+12\right)$ .
$8{a}^{3}+{a}^{2}-2a+12$
Find the sum of
$\left({b}^{2}+5b+13\right)$ and
$\left(4{b}^{3}-6\right)$ .
Find the difference of
$\left({w}^{2}+w-42\right)$ and
$\left({w}^{2}-10w+24\right)$ .
$11w-64$
Find the difference of
$\left({z}^{2}-3z-18\right)$ and
$\left({z}^{2}+5z-20\right)$ .
Find the difference of
$\left({c}^{2}+4c-33\right)$ and
$\left({c}^{2}-8c+12\right)$ .
$12c-45$
Find the difference of
$\left({t}^{2}-5t-15\right)$ and
$\left({t}^{2}+4t-17\right)$ .
$\left(7{x}^{2}-2xy+6{y}^{2}\right)+\left(3{x}^{2}-5xy\right)$
$10{x}^{2}-7xy+6{y}^{2}$
$\left(\mathrm{-5}{x}^{2}-4xy-3{y}^{2}\right)+\left(2{x}^{2}-7xy\right)$
$\left(7{m}^{2}+mn-8{n}^{2}\right)+\left(3{m}^{2}+2mn\right)$
$10{m}^{2}+3mn-8{n}^{2}$
$\left(2{r}^{2}-3rs-2{s}^{2}\right)+\left(5{r}^{2}-3rs\right)$
$\left({a}^{2}-{b}^{2}\right)-\left({a}^{2}+3ab-4{b}^{2}\right)$
$\mathrm{-3}ab+3{b}^{2}$
$\left({m}^{2}+2{n}^{2}\right)-\left({m}^{2}-8mn-{n}^{2}\right)$
$\left({u}^{2}-{v}^{2}\right)-\left({u}^{2}-4uv-3{v}^{2}\right)$
$4uv+2{v}^{2}$
$\left({j}^{2}-{k}^{2}\right)-\left({j}^{2}-8jk-5{k}^{2}\right)$
$\left({p}^{3}-3{p}^{2}q\right)+\left(2p{q}^{2}+4{q}^{3}\right)$ $-\left(3{p}^{2}q+p{q}^{2}\right)$
${p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}$
$\left({a}^{3}-2{a}^{2}b\right)+\left(a{b}^{2}+{b}^{3}\right)$ $-\left(3{a}^{2}b+4a{b}^{2}\right)$
$\left({x}^{3}-{x}^{2}y\right)-\left(4x{y}^{2}-{y}^{3}\right)$ $+\left(3{x}^{2}y-x{y}^{2}\right)$
${x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}$
$\left({x}^{3}-2{x}^{2}y\right)-\left(x{y}^{2}-3{y}^{3}\right)$ $-\left({x}^{2}y-4x{y}^{2}\right)$
Evaluate a Polynomial for a Given Value
In the following exercises, evaluate each polynomial for the given value.
Evaluate $8{y}^{2}-3y+2$ when:
ⓐ
$y=5$
ⓑ
$y=\mathrm{-2}$
ⓒ
$y=0$
ⓐ 187 ⓑ 46 ⓒ 2
Evaluate $5{y}^{2}-y-7$ when:
ⓐ
$y=\mathrm{-4}$
ⓑ
$y=1$
ⓒ
$y=0$
Evaluate $4-36x$ when:
ⓐ
$x=3$
ⓑ
$x=0$
ⓒ
$x=\mathrm{-1}$
ⓐ −104 ⓑ 4 ⓒ 40
Evaluate $16-36{x}^{2}$ when:
ⓐ
$x=\mathrm{-1}$
ⓑ
$x=0$
ⓒ
$x=2$
A painter drops a brush from a platform 75 feet high. The polynomial $\mathrm{-16}{t}^{2}+75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t=2$ seconds.
11
A girl drops a ball off a cliff into the ocean. The polynomial $\mathrm{-16}{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall cliff. Find the height after $t=2$ seconds.
A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $\mathrm{-4}{p}^{2}+420p.$ Find the revenue received when $p=60$ dollars.
$10,800
A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial $\mathrm{-4}{p}^{2}+420p.$ Find the revenue received when $p=90$ dollars.
Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of $x$ miles per hour is given by the polynomial $-\frac{1}{150}{x}^{2}+\frac{1}{3}x$ . Find the fuel efficiency when $x=30\phantom{\rule{0.2em}{0ex}}\text{mph}$ .
4
Stopping Distance The number of feet it takes for a car traveling at $x$ miles per hour to stop on dry, level concrete is given by the polynomial $0.06{x}^{2}+1.1x$ . Find the stopping distance when $x=40\phantom{\rule{0.2em}{0ex}}\text{mph}$ .
Rental Cost The cost to rent a rug cleaner for $d$ days is given by the polynomial $5.50d+25$ . Find the cost to rent the cleaner for 6 days.
$58
Height of Projectile The height (in feet) of an object projected upward is given by the polynomial $\mathrm{-16}{t}^{2}+60t+90$ where $t$ represents time in seconds. Find the height after $t=2.5$ seconds.
Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial $\frac{9}{5}c+32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c=65\text{\xb0}.$
149
Using your own words, explain the difference between a monomial, a binomial, and a trinomial.
Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.
Answers will vary.
Ariana thinks the sum $6{y}^{2}+5{y}^{4}$ is $11{y}^{6}$ . What is wrong with her reasoning?
Jonathan thinks that $\frac{1}{3}$ and $\frac{1}{x}$ are both monomials. What is wrong with his reasoning?
Answers will vary.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.
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