5.1 Add and Subtract Polynomials

Learning objectives

Learning objectives

• Determine the degree of polynomials

• Add and subtract polynomials

• Evaluate a polynomial function for a given value

• Add and subtract polynomial functions

Determine the Degree of Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form $a{x}^{m},$ where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomials in one variable are $2x,5y,17z,$ and $4{y}^{2}$. Monomials can also have more than one variable such as $5abc$ and $-4{a}^{2}{b}^{3}{c}^{2}.$

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Here are some examples of polynomials.

Polynomial · $y+1$ · $4{a}^{2}-7ab+2{b}^{2}$ · $4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1$ Monomial · 14 · $8{y}^{2}$ · $-9{x}^{3}{y}^{5}$ · $-13{a}^{3}{b}^{2}c$ Binomial · $a+7b$ · $4{x}^{2}-{y}^{2}$ · ${y}^{2}-16$ · $3{p}^{3}q-9{p}^{2}q$ Trinomial · ${x}^{2}-7x+12$ · $9{m}^{2}+2mn-8{n}^{2}$ · $6{k}^{4}-{k}^{3}+8k$ · ${z}^{4}+3{z}^{2}-1$

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Let's start by looking at a monomial. The monomial $8a{b}^{2}$ has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, $1+2,$ is 3 so the degree is 3.

Here are some additional examples.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

Example 5.1

Solution

Polynomial · Number of terms · Type · Degree of terms · Degree of polynomial ⓐ · $7{y}^{2}-5y+3$ · 3 · Trinomial · 2, 1, 0 · 2 ⓑ · $-2{a}^{4}{b}^{2}$ · 1 · Monomial · 6 · 6 ⓒ · $3{x}^{5}-4{x}^{3}-6{x}^{2}+x-8$ · 5 · Polynomial · 5, 3, 2, 1, 0 · 5 ⓓ · $2y-8x{y}^{3}$ · 2 · Binomial · 1, 4 · 4 ⓔ · 15 · 1 · Monomial · 0 · 0

Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

Example 5.2

Solution

ⓐ $\begin{array}{cccc} & & & \,25{y}^{2}+15{y}^{2} \\ \mathrm{Combine like terms.} & & & \,40{y}^{2} \\ \end{array}$

ⓑ $\begin{array}{cccc} & & & \,16p{q}^{3}-(-7p{q}^{3}) \\ \mathrm{Combine like terms.} & & & \,23p{q}^{3} \\ \end{array}$

Remember that like terms must have the same variables with the same exponents.

Example 5.3

Solution

ⓐ

$\,{a}^{2}+7{b}^{2}-6{a}^{2}$

Combine like terms. · $\,-5{a}^{2}+7{b}^{2}$

ⓑ

${u}^{2}v+5{u}^{2}-3{v}^{2}$

There are no like terms to combine.In this case, the polynomial is unchanged. · ${u}^{2}v+5{u}^{2}-3{v}^{2}$

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example 5.4

Solution

Identify like terms. · $(\underset{\underline{\hspace{2em}}}{\underset{\underline{\hspace{2em}}}{7{y}^{2}}}-\underset{\mathrm{\_\_\_}}{2y}+9)+(\underset{\underline{\hspace{2em}}}{\underset{\underline{\hspace{2em}}}{4{y}^{2}}}-\underset{\mathrm{\_\_\_}}{8y}-7)$ Rewrite without the parentheses,rearranging to get the like terms together. · $\underset{\mathrm{\_\_\_\_\_\_\_\_\_}}{\underset{\mathrm{\_\_\_\_\_\_\_\_\_}}{7{y}^{2}+4{y}^{2}}}-\underset{\mathrm{\_\_\_\_\_\_\_}}{2y-8y}+9-7$ Combine like terms. · $11{y}^{2}-10y+2$

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Example 5.5

Solution

$(9{w}^{2}-7w+5)-(2{w}^{2}-4)$

Distribute and identify like terms. · $\underset{\underline{\hspace{2em}}}{\underset{\underline{\hspace{2em}}}{9{w}^{2}}}-\underset{\mathrm{\_\_\_}}{7w}+5-\underset{\underline{\hspace{2em}}}{\underset{\underline{\hspace{2em}}}{2{w}^{2}}}+4$ Rearrange the terms. · $\underset{\mathrm{\_\_\_\_\_\_\_\_\_\_}}{\underset{\mathrm{\_\_\_\_\_\_\_\_\_\_}}{9{w}^{2}-2{w}^{2}}}-\underset{\mathrm{\_\_\_}}{7w}+5+4$ Combine like terms. · $7{w}^{2}-7w+9$

To subtract $a$ from $b,$ we write it as $b-a,$ placing the $b$ first.

Example 5.6

Solution

$({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})$

Distribute. · ${p}^{2}+{q}^{2}-{p}^{2}-10pq+2{q}^{2}$ Rearrange the terms, to put like terms together. · ${p}^{2}-{p}^{2}-10pq+{q}^{2}+2{q}^{2}$ Combine like terms. · $-10pq+3{q}^{2}$

Example 5.7

Solution

$({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv)$

Distribute. · ${u}^{2}-6uv+5{v}^{2}+3{u}^{2}+2uv$ Rearrange the terms to put like terms together. · ${u}^{2}+3{u}^{2}-6uv+2uv+5{v}^{2}$ Combine like terms. · $4{u}^{2}-4uv+5{v}^{2}$

When we add and subtract more than two polynomials, the process is the same.

Example 5.8

Solution

$({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2})$

Distribute. · ${a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}$ Rewrite without the parentheses,rearranging to get the like terms together. · ${a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}$ Combine like terms. · ${a}^{3}-{b}^{3}$

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, $f(x)={x}^{2}+5x+6$ and $g(x)=3x-4$ are polynomial functions, because ${x}^{2}+5x+6$ and $3x-4$ are polynomials.

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of $f(x)$ for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example 5.9

Solution

ⓐ

ⓑ

ⓒ

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Example 5.10

Solution

$h(t)=-16{t}^{2}+250$

To find $h(2),$ substitute $t=2.$ · $h(2)=-16{(2)}^{2}+250$ Simplify. · $h(2)=-16\cdot{}4+250$ Simplify. · $h(2)=-64+250$ Simplify. · $h(2)=186$

After 2 seconds the height of the ball is 186 feet.

Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

Example 5.11

Solution

ⓐ

ⓑ In part (a) we found $(f+g)(x)$ and now are asked to find $(f+g)(3).$

$(f+g)(x)=4{x}^{2}-9x+4$

To find $(f+g)(3),$ substitute $x=3.$ · $(f+g)(3)=4{(3)}^{2}-9\cdot{}3+4$

$(f+g)(3)=4\cdot{}9-9\cdot{}3+4$

$(f+g)(3)=36-27+4$

Notice that we could have found $(f+g)(3)$ by first finding the values of $f(3)$ and $g(3)$ separately and then adding the results.

ⓒ

ⓓ

Section 5.1 Exercises

Section 5.1 Exercises

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

In the following exercises, add the polynomials.

In the following exercises, subtract the polynomials.

In the following exercises, subtract the polynomials.

In the following exercises, find the difference of the polynomials.

In the following exercises, add the polynomials.

In the following exercises, add or subtract the polynomials.

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

In the following exercises, find the height for each polynomial function.

Add and Subtract Polynomial Functions

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

Writing Exercises

ⓐ 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. Whom 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.

1. Add: ⓐ $8{y}^{2}+3{z}^{2}-3{y}^{2}$ ⓑ ${m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}.$

2. Add: ⓐ $3{m}^{2}+{n}^{2}-7{m}^{2}$ ⓑ $p{q}^{2}-6p-5{q}^{2}.$

3. Find the sum: $(7{x}^{2}-4x+5)+({x}^{2}-7x+3).$

4. Find the sum: $(14{y}^{2}+6y-4)+(3{y}^{2}+8y+5).$

1. Find the difference: $(8{x}^{2}+3x-19)-(7{x}^{2}-14).$

2. Find the difference: $(9{b}^{2}-5b-4)-(3{b}^{2}-5b-7).$

3. Subtract $({a}^{2}+5ab-6{b}^{2})$ from $({a}^{2}+{b}^{2}).$

4. Subtract $({m}^{2}-7mn-3{n}^{2})$ from $({m}^{2}+{n}^{2}).$

1. Find the sum: $(3{x}^{2}-4xy+5{y}^{2})+(2{x}^{2}-xy).$

2. Find the sum: $(2{x}^{2}-3xy-2{y}^{2})+(5{x}^{2}-3xy).$

3. Simplify: $({x}^{3}-{x}^{2}y)-(x{y}^{2}+{y}^{3})+({x}^{2}y+x{y}^{2}).$

4. Simplify: $({p}^{3}-{p}^{2}q)+(p{q}^{2}+{q}^{3})-({p}^{2}q+p{q}^{2}).$

1. For the function $f(x)=3{x}^{2}+2x-15,$ find ⓐ $f(3)$ ⓑ $f(-5)$ ⓒ $f(0).$

2. For the function $g(x)=5{x}^{2}-x-4,$ find ⓐ $g(-2)$ ⓑ $g(-1)$ ⓒ $g(0).$

3. The polynomial function $h(t)=-16{t}^{2}+150$ gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after $t=0$ seconds (the initial height of the object).

4. The polynomial function $h(t)=-16{t}^{2}+175$ gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after $t=3$ seconds.

1. For functions $f(x)=2{x}^{2}-4x+3$ and $g(x)={x}^{2}-2x-6,$ find: ⓐ $(f+g)(x)$ ⓑ $(f+g)(3)$ ⓒ $(f-g)(x)$ ⓓ $(f-g)(-2).$

2. For functions $f(x)=5{x}^{2}-4x-1$ and $g(x)={x}^{2}+3x+8,$ find ⓐ $(f+g)(x)$ ⓑ $(f+g)(3)$ ⓒ $(f-g)(x)$ ⓓ $(f-g)(-2).$

3. ⓐ $47{x}^{5}-17{x}^{2}{y}^{3}+{y}^{2}$ⓑ $5{c}^{3}+11{c}^{2}-c-8$ⓒ $\frac{5}{9}ab+\frac{1}{3}b$ⓓ 4ⓔ $4pq+17$

4. ⓐ $8y-5x$ⓑ ${y}^{2}-5yz-6{z}^{2}$ⓒ ${y}^{3}-8{y}^{2}+2y-16$ⓓ $81a{b}^{4}-24{a}^{2}{b}^{2}+3b$ⓔ $-18$

1. ⓐ $5{a}^{2}+12ab-7{b}^{2}$ⓑ $18x{y}^{2}z$ⓒ $5x+2$ⓓ ${y}^{3}-8{y}^{2}+2y-16$ⓔ $-24$

2. ⓐ $14s-29t$ⓑ ${z}^{2}-5z-6$ⓒ ${y}^{3}-8{y}^{2}z+2y{z}^{2}-16{z}^{3}$ⓓ $23a{b}^{2}-14$ⓔ $-3$

3. ⓐ ${\,\mathrm{7x}}^{2}+5{x}^{2}$ⓑ $\,\mathrm{4a}-9a$

4. ⓐ $-12w+18w$ⓑ $7{x}^{2}y-(-12{x}^{2}y)$

1. ${\,\mathrm{7x}}^{2}+5{x}^{2}+\,\mathrm{4a}-9a$

2. $-12w+18w+7{x}^{2}y-(-12{x}^{2}y)$

3. ⓐ $-5b-17b$ⓑ $3xy-(-8xy)+5xy$

4. ⓐ $\,\mathrm{12}a+5b-22a$ⓑ $p{q}^{2}-4p-3{q}^{2}$

1. ⓐ $2{a}^{2}+{b}^{2}-6{a}^{2}$ⓑ ${x}^{2}y-3x+7x{y}^{2}$

2. ⓐ $x{y}^{2}-5x-5{y}^{2}$ⓑ $\,\mathrm{19y}+5z$

3. $\mathrm{14x}-3y-13x+{a}^{2}b-4a-5a{b}^{2}$

4. $5{u}^{2}+4{v}^{2}-6{u}^{2}+\,\mathrm{12a}+8b$

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

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

3. $(5{y}^{2}+12y+4)+(6{y}^{2}-8y+7)$

4. $({x}^{2}+6x+8)+(-4{x}^{2}+11x-9)$

1. $(8{x}^{2}-5x+2)+(3{x}^{2}+3)$

2. $(5{a}^{2}+8)+({a}^{2}-4a-9)$

3. $(4{m}^{2}-6m-3)-(2{m}^{2}+m-7)$

4. $({a}^{2}+8a+5)-({a}^{2}-3a+2)$

1. $(12{s}^{2}-15s)-(s-9)$

2. Subtract $(9{x}^{2}+2)$ from $(12{x}^{2}-x+6)$

3. Subtract $(7{w}^{2}-4w+2)$ from $(8{w}^{2}-w+6)$

4. Find the difference of $({w}^{2}+w-42)$ and $({w}^{2}-10w+24)$

1. $(7{x}^{2}-2xy+6{y}^{2})+(3{x}^{2}-5xy)$

2. $(7{m}^{2}+mn-8{n}^{2})+(3{m}^{2}+2mn)$

3. $({a}^{2}-{b}^{2})-({a}^{2}+3ab-4{b}^{2})$

4. $({p}^{3}-3{p}^{2}q)+(2p{q}^{2}+4{q}^{3})-(3{p}^{2}q+p{q}^{2})$

1. $({x}^{3}-{x}^{2}y)-(4x{y}^{2}-{y}^{3})+(3{x}^{2}y-x{y}^{2})$

2. For the function $f(x)=8{x}^{2}-3x+2,$ find:ⓐ $f(5)$ ⓑ $f(-2)$ ⓒ $f(0)$

3. For the function $g(x)=4-36x,$ find:ⓐ $g(3)$ ⓑ $g(0)$ ⓒ $g(-1)$

4. A painter drops a brush from a platform 75 feet high. The polynomial function $h(t)=-16{t}^{2}+75$ gives the height of the brush t seconds after it was dropped. Find the height after $t=2$ seconds.

1. 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 function $R(p)=-4{p}^{2}+420p.$ Find the revenue received when $p=60$ dollars.

2. The polynomial $C(x)=6{x}^{2}+90x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with $x=4$ feet.

3. $f(x)=2{x}^{2}-4x+1$ and $g(x)=5{x}^{2}+8x+3$

4. $f(x)=3{x}^{3}-{x}^{2}-2x+3$ and $g(x)=3{x}^{3}-7x$

1. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $-5$ ⓑ $8{y}^{3}-7{y}^{2}-y-3$ ⓒ $-3{x}^{2}y-5xy+9x{y}^{3}$ ⓓ $81{m}^{2}-4{n}^{2}$ ⓔ $-3{x}^{6}{y}^{3}z$

2. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $64{k}^{3}-8$ ⓑ $9{m}^{3}+4{m}^{2}-2$ ⓒ $\frac{5}{6}$ ⓓ $8{a}^{4}-7{a}^{3}b-6{a}^{2}{b}^{2}-4a{b}^{3}+7{b}^{4}$ ⓔ $-{p}^{4}{q}^{3}$

3. Add or subtract: ⓐ $12{q}^{2}+9{q}^{2}$ ⓑ $8m{n}^{3}-(-5m{n}^{3}).$

4. Add or subtract: ⓐ $-15{c}^{2}+8{c}^{2}$ ⓑ $-15{y}^{2}{z}^{3}-(-5{y}^{2}{z}^{3}).$

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

2. Ariana thinks the sum $6{y}^{2}+5{y}^{4}$ is $11{y}^{6}.$ What is wrong with her reasoning?