# 6.1 Add and subtract polynomials  (Page 2/12)

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## Practice makes perfect

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

${x}^{2}-{y}^{2}$
$-13{c}^{4}$
${x}^{2}+5x-7$
${x}^{2}{y}^{2}-2xy+8$
19

$8-3x$
${z}^{2}-5z-6$
${y}^{3}-8{y}^{2}+2y-16$
$81{b}^{5}-24{b}^{3}+1$
$-18$

binomial trinomial polynomial trinomial monomial

$11{y}^{2}$
$-73$
$6{x}^{2}-3xy+4x-2y+{y}^{2}$
$4y+17$
$5{c}^{3}+11{c}^{2}-c-8$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

$6{a}^{2}+12a+14$
$18x{y}^{2}z$
$5x+2$
${y}^{3}-8{y}^{2}+2y-16$
$-24$

2 4 1 3 0

$9{y}^{3}-10{y}^{2}+2y-6$
$-12{p}^{4}$
${a}^{2}+9a+18$
$20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30$
17

$14-29x$
${z}^{2}-5z-6$
${y}^{3}-8{y}^{2}+2y-16$
$23a{b}^{2}-14$
$-3$

1 2 3 3 0

$62{y}^{2}$
15
$6{x}^{2}-3xy+4x-2y+{y}^{2}$
$10-9x$
${m}^{4}+4{m}^{3}+6{m}^{2}+4m+1$

In the following exercises, add or subtract the monomials.

${\phantom{\rule{0.2em}{0ex}}\text{7x}}^{2}+5{x}^{2}$

$12{x}^{2}$

${\phantom{\rule{0.2em}{0ex}}\text{4y}}^{3}+6{y}^{3}$

$-12w+18w$

$6w$

$-3m+9m$

$\text{4a}-9a$

$-5a$

$\text{−}y-5y$

$28x-\left(-12x\right)$

$40x$

$13z-\left(-4z\right)$

$-5b-17b$

$-22b$

$-10x-35x$

$12a+5b-22a$

$\text{−10a}+5b$

$\text{14x}-3y-13x$

$2{a}^{2}+{b}^{2}-6{a}^{2}$

$-4{a}^{2}+{b}^{2}$

$5{u}^{2}+4{v}^{2}-6{u}^{2}$

$x{y}^{2}-5x-5{y}^{2}$

$x{y}^{2}-5x-5{y}^{2}$

$p{q}^{2}-4p-3{q}^{2}$

${a}^{2}b-4a-5a{b}^{2}$

${a}^{2}b-4a-5a{b}^{2}$

${x}^{2}y-3x+7x{y}^{2}$

$\text{12a}+8b$

$\text{12a}+8b$

$\text{19y}+5z$

Add: $4a,-3b,-8a$

$-4a-3b$

Add: $\phantom{\rule{0.2em}{0ex}}\text{4x},3y,-3x$

Subtract $5{x}^{6}\text{from}-12{x}^{6}$ .

$-17{x}^{6}$

Subtract $2{p}^{4}\text{from}-7{p}^{4}$ .

In the following exercises, add or subtract the polynomials.

$\left(5{y}^{2}+12y+4\right)+\left(6{y}^{2}-8y+7\right)$

$11{y}^{2}+4y+11$

$\left(4{y}^{2}+10y+3\right)+\left(8{y}^{2}-6y+5\right)$

$\left({x}^{2}+6x+8\right)+\left(-4{x}^{2}+11x-9\right)$

$-3{x}^{2}+17x-1$

$\left({y}^{2}+9y+4\right)+\left(-2{y}^{2}-5y-1\right)$

$\left(8{x}^{2}-5x+2\right)+\left(3{x}^{2}+3\right)$

$11{x}^{2}-5x+5$

$\left(7{x}^{2}-9x+2\right)+\left(6{x}^{2}-4\right)$

$\left(5{a}^{2}+8\right)+\left({a}^{2}-4a-9\right)$

$6{a}^{2}-4a-1$

$\left({p}^{2}-6p-18\right)+\left(2{p}^{2}+11\right)$

$\left(4{m}^{2}-6m-3\right)-\left(2{m}^{2}+m-7\right)$

$2{m}^{2}-7m+4$

$\left(3{b}^{2}-4b+1\right)-\left(5{b}^{2}-b-2\right)$

$\left({a}^{2}+8a+5\right)-\left({a}^{2}-3a+2\right)$

$5a+3$

$\left({b}^{2}-7b+5\right)-\left({b}^{2}-2b+9\right)$

$\left(12{s}^{2}-15s\right)-\left(s-9\right)$

$12{s}^{2}-14s+9$

$\left(10{r}^{2}-20r\right)-\left(r-8\right)$

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
$\left({t}^{2}+4t-17\right)$ .

$\left(7{x}^{2}-2xy+6{y}^{2}\right)+\left(3{x}^{2}-5xy\right)$

$10{x}^{2}-7xy+6{y}^{2}$

$\left(-5{x}^{2}-4xy-3{y}^{2}\right)+\left(2{x}^{2}-7xy\right)$

$\left(7{m}^{2}+mn-8{n}^{2}\right)+\left(3{m}^{2}+2mn\right)$

$10{m}^{2}+3mn-8{n}^{2}$

$\left(2{r}^{2}-3rs-2{s}^{2}\right)+\left(5{r}^{2}-3rs\right)$

$\left({a}^{2}-{b}^{2}\right)-\left({a}^{2}+3ab-4{b}^{2}\right)$

$-3ab+3{b}^{2}$

$\left({m}^{2}+2{n}^{2}\right)-\left({m}^{2}-8mn-{n}^{2}\right)$

$\left({u}^{2}-{v}^{2}\right)-\left({u}^{2}-4uv-3{v}^{2}\right)$

$4uv+2{v}^{2}$

$\left({j}^{2}-{k}^{2}\right)-\left({j}^{2}-8jk-5{k}^{2}\right)$

$\left({p}^{3}-3{p}^{2}q\right)+\left(2p{q}^{2}+4{q}^{3}\right)$ $-\left(3{p}^{2}q+p{q}^{2}\right)$

${p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}$

$\left({a}^{3}-2{a}^{2}b\right)+\left(a{b}^{2}+{b}^{3}\right)$ $-\left(3{a}^{2}b+4a{b}^{2}\right)$

$\left({x}^{3}-{x}^{2}y\right)-\left(4x{y}^{2}-{y}^{3}\right)$ $+\left(3{x}^{2}y-x{y}^{2}\right)$

${x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}$

$\left({x}^{3}-2{x}^{2}y\right)-\left(x{y}^{2}-3{y}^{3}\right)$ $-\left({x}^{2}y-4x{y}^{2}\right)$

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:

$y=5$
$y=-2$
$y=0$

187 46 2

Evaluate $5{y}^{2}-y-7$ when:

$y=-4$
$y=1$
$y=0$

Evaluate $4-36x$ when:

$x=3$
$x=0$
$x=-1$

−104 4 40

Evaluate $16-36{x}^{2}$ when:

$x=-1$
$x=0$
$x=2$

A painter drops a brush from a platform 75 feet high. The polynomial $-16{t}^{2}+75$ 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 $-16{t}^{2}+250$ 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 $-4{p}^{2}+420p.$ 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 $-4{p}^{2}+420p.$ Find the revenue received when $p=90$ dollars. ## Everyday math 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 $-16{t}^{2}+60t+90$ 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 $c=65\text{°}.$

149

## Writing exercises

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.

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?

## Self check

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.

The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold? Mum Reply A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used? Edi Reply June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Jesus Reply Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself? Ronald Reply Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles. Dojzae Reply LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution? Xona Reply 5% Michael hey everyone how to do algebra The Reply Felecia answer 1.5 hours before he reaches her Adriana Reply I would like to solve the problem -6/2x rachel Reply 12x Andrew how Christian Does the x represent a number or does it need to be graphed ? latonya -3/x Venugopal -3x is correct Atul Arnold invested$64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? Stephanie Reply Tickets for the community fair cost$12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of$2204. How many adult tickets and how many child tickets were sold?
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla