# 10.1 Add and subtract polynomials  (Page 2/10)

 Page 3 / 10

Evaluate $3{x}^{2}-9x+7$ when

1. $\phantom{\rule{0.2em}{0ex}}x=3$
2. $\phantom{\rule{0.2em}{0ex}}x=-1$

## Solution

 ⓐ $x=3$ $3{x}^{2}-9x+7$ Substitute 3 for $x$ $3{\left(3\right)}^{2}-9\left(3\right)+7$ Simplify the expression with the exponent. $3·9-9\left(3\right)+7$ Multiply. $27-27+7$ Simplify. $7$
 ⓑ $x=-1$ $3{x}^{2}-9x+7$ Substitute −1 for $x$ $3{\left(-1\right)}^{2}-9\left(-1\right)+7$ Simplify the expression with the exponent. $3·1-9\left(-1\right)+7$ Multiply. $3+9+7$ Simplify. $19$

Evaluate: $2{x}^{2}+4x-3$ when

1. $\phantom{\rule{0.2em}{0ex}}x=2$
2. $\phantom{\rule{0.2em}{0ex}}x=-3$

1. 13
2. 3

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

1. $\phantom{\rule{0.2em}{0ex}}y=-4$
2. $\phantom{\rule{0.2em}{0ex}}y=0$

1. 114
2. −2

The polynomial $-16{t}^{2}+300$ gives the height of an object $t$ seconds after it is dropped from a $300$ foot tall bridge. Find the height after $t=3$ seconds.

## Solution

 Substitute 3 for $t$ Simplify the expression with the exponent. Multiply. Simplify.

The polynomial $-8{t}^{2}+24t+4$ gives the height, in feet, of a ball $t$ seconds after it is tossed into the air, from an initial height of $4$ feet. Find the height after $t=3$ seconds.

4 feet

The polynomial $-8{t}^{2}+24t+4$ gives the height, in feet, of a ball $x$ seconds after it is tossed into the air, from an initial height of $4$ feet. Find the height after $t=2$ seconds.

20 feet

## Practice makes perfect

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial.

$5x+2$

binomial

${z}^{2}-5z-6$

${a}^{2}+9a+18$

trinomial

$-12{p}^{4}$

${y}^{3}-8{y}^{2}+2y-16$

polynomial

$10-9x$

$23{y}^{2}$

monomial

${m}^{4}+4{m}^{3}+6{m}^{2}+4m+1$

Determine the Degree of Polynomials

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

$8{a}^{5}-2{a}^{3}+1$

5

$5{c}^{3}+11{c}^{2}-c-8$

$3x-12$

1

$4y+17$

$-13$

0

$-22$

In the following exercises, add or subtract the monomials.

${\text{6x}}^{2}+\phantom{\rule{0.2em}{0ex}}9{x}^{2}$

15 x 2

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

$-12u\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}4u$

−8 u

$-3m\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}9m$

$5a\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}7b$

5 a + 7 b

$8y\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}6z$

Add: $\text{}4a\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}-3b,\phantom{\rule{0.2em}{0ex}}-8a$

−4 a −3 b

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

$18x-2x$

16 x

$13a-3a$

Subtract $5{x}^{6}\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}-12{x}^{6}$

−17 x 6

Subtract $2{p}^{4}\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}-7{p}^{4}$

In the following exercises, add or subtract the polynomials.

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

12 y 2 + 4 y + 8

$\left(7{x}^{2}-9x+2\right)+\left(6{x}^{2}-4x+3\right)$

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

−3 x 2 + 17 x − 1

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

$\left(3{a}^{2}+7\right)+\left({a}^{2}-7a-18\right)$

4 a 2 − 7 a − 11

$\left({p}^{2}-5p-11\right)+\left(3{p}^{2}+9\right)$

$\left(6{m}^{2}-9m-3\right)-\left(2{m}^{2}+m-5\right)$

4 m 2 − 10 m + 2

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

$\left({z}^{2}+8z+9\right)-\left({z}^{2}-3z+1\right)$

11 z + 8

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

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

12 s 2 − 16 s + 9

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

Find the sum of $\left(2{p}^{3}-8\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left({p}^{2}+9p+18\right)$

2 p 3 + p 2 + 9 p + 10

Find the sum of $\left({q}^{2}+4q+13\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(7{q}^{3}-3\right)$

Subtract $\left(7{x}^{2}-4x+2\right)\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}\left(8{x}^{2}-x+6\right)$

x 2 + 3 x + 4

Subtract $\left(5{x}^{2}-x+12\right)\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}\left(9{x}^{2}-6x-20\right)$

Find the difference of $\left({w}^{2}+w-42\right)$ and $\left({w}^{2}-10w+24\right)$

11 w − 66

Find the difference of $\left({z}^{2}-3z-18\right)$ and $\left({z}^{2}+5z-20\right)$

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}8{y}^{2}-3y+2$

1. $\phantom{\rule{0.2em}{0ex}}y=5$
2. $\phantom{\rule{0.2em}{0ex}}y=-2$
3. $\phantom{\rule{0.2em}{0ex}}y=0$

1. 187
2. 40
3. 2

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}5{y}^{2}-y-7\phantom{\rule{0.2em}{0ex}}\text{when:}$

1. $\phantom{\rule{0.2em}{0ex}}y=-4\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}y=1$
3. $y=0$

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}4-36x\phantom{\rule{0.2em}{0ex}}\text{when:}$

1. $\phantom{\rule{0.2em}{0ex}}x=3$
2. $\phantom{\rule{0.2em}{0ex}}x=0$
3. $x=-1$

1. −104
2. 4
3. 40

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}16-36{x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when:}$

1. $\phantom{\rule{0.2em}{0ex}}x=-1\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}x=0$
3. $x=2$

A window washer drops a squeegee from a platform $275$ feet high. The polynomial $-16{t}^{2}+275$ gives the height of the squeegee $t$ seconds after it was dropped. Find the height after $t=4$ seconds.

19 feet

A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p dollars each is given by the polynomial $-5{p}^{2}+350p.$ Find the revenue received when $p=50$ dollars.

## Everyday math

Fuel Efficiency The fuel efficiency (in miles per gallon) of a bus going at a speed of $x$ miles per hour is given by the polynomial $-\frac{1}{160}\phantom{\rule{0.1em}{0ex}}{x}^{2}+\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x.$ Find the fuel efficiency when $x=40\phantom{\rule{0.2em}{0ex}}\text{mph.}$

10 mpg

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 $x=60\phantom{\rule{0.2em}{0ex}}\text{mph.}$

## Writing exercises

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Eloise thinks the sum $5{x}^{2}+3{x}^{4}$ is $8{x}^{6}.$ What is wrong with her 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.

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