# 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.

how many typos can we find...?
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9