6.1 Add and Subtract Polynomials

Learning objectives

Learning objectives

• Identify polynomials, monomials, binomials, and trinomials

• Determine the degree of polynomials

• Add and subtract monomials

• Add and subtract polynomials

• Evaluate a polynomial for a given value

Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. The constant is called a coefficient. 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. Some examples of monomial are $8,-2{x}^{2},4{y}^{3},\,\mathrm{and}\,11{z}^{7}$.

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.

$\begin{array}{lcclcclcclccc} \mathrm{Polynomial} & & & \,b+1 & & & \,4{y}^{2}-7y+2 & & & \,4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1 & & & \\ \mathrm{Monomial} & & & \,14 & & & \,8{y}^{2} & & & \,-9{x}^{3}{y}^{5} & & & \,-13 \\ \mathrm{Binomial} & & & \,a+7 & & & \,4b-5 & & & \,{y}^{2}-16 & & & \,3{x}^{3}-9{x}^{2} \\ \mathrm{Trinomial} & & & \,{x}^{2}-7x+12 & & & \,9{y}^{2}+2y-8 & & & \,6{m}^{4}-{m}^{3}+8m & & & \,{z}^{4}+3{z}^{2}-1 \\ \end{array}$

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.

Example 6.1

Solution

Determine the Degree of 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—it has no variable.

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.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example 6.2

Solution

Add and Subtract Monomials

You 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 coefficient.

Example 6.3

Solution

Example 6.4

Solution

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

Example 6.5

Solution

Example 6.6

Solution

Add and Subtract Polynomials

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 6.7

Solution

Example 6.8

Solution

Example 6.9

Solution

Example 6.10

Solution

Example 6.11

Solution

Example 6.12

Solution

Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

Example 6.13

Solution

Example 6.14

Solution

Example 6.15

Solution

Section 6.1 Exercises

Section 6.1 Exercises

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.

Determine the Degree of Polynomials

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

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Evaluate a Polynomial for a Given Value

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

Everyday Math

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. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

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

2. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ $27{z}^{3}-8$ ⓑ $12{m}^{3}-5{m}^{2}-2m$ ⓒ $\frac{5}{6}$ ⓓ $8{x}^{4}-7{x}^{2}-6x-5$ ⓔ $-{n}^{4}$

3. Find the degree of the following polynomials:

ⓐ $-15b$ ⓑ $10{z}^{4}+4{z}^{2}-5$ ⓒ $12{c}^{5}{d}^{4}+9{c}^{3}{d}^{9}-7$ ⓓ $3{x}^{2}y-4x$ ⓔ $-9$

4. Find the degree of the following polynomials:

ⓐ $52$ ⓑ ${a}^{4}b-17{a}^{4}$ ⓒ $5x+6y+2z$ ⓓ $3{x}^{2}-5x+7$ ⓔ $-{a}^{3}$

5. Add: $12{q}^{2}+9{q}^{2}.$

1. Add: $-15{c}^{2}+8{c}^{2}.$

2. Subtract: $8m-(-5m).$

3. Subtract: $-15{z}^{3}-(-5{z}^{3}).$

4. Add: $8{y}^{2}+3{z}^{2}-3{y}^{2}.$

5. Add: $3{m}^{2}+{n}^{2}-7{m}^{2}.$

1. Simplify: ${m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}.$

2. Simplify: $p{q}^{2}-6p-5{q}^{2}.$

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

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

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

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

2. Subtract: $(5{z}^{2}-6z-2)$ from $(7{z}^{2}+6z-4)$.

3. Subtract: $({x}^{2}-5x-8)$ from $(6{x}^{2}+9x-1)$.

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

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

1. Find the difference: $({a}^{2}+{b}^{2})-({a}^{2}+5ab-6{b}^{2})$.

2. Find the difference: $({m}^{2}+{n}^{2})-({m}^{2}-7mn-3{n}^{2})$.

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

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

5. Evaluate: $3{x}^{2}+2x-15$ when

1. ⓐ $x=3$

2. ⓑ $x=-5$

3. ⓒ $x=0$

1. Evaluate: $5{z}^{2}-z-4$ when

1. ⓐ $z=-2$

2. ⓑ $z=0$

3. ⓒ $z=2$

2. The polynomial $-16{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t=0$ seconds.

3. The polynomial $-16{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t=3$ seconds.

4. The polynomial $6{x}^{2}+15xy$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x=6$ feet and $y=4$ feet.

5. The polynomial $6{x}^{2}+15xy$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x=5$ feet and $y=8$ feet.

1. ⓐ $81{b}^{5}-24{b}^{3}+1$ ⓑ $5{c}^{3}+11{c}^{2}-c-8$ ⓒ $\frac{14}{15}y+\frac{1}{7}$ ⓓ 5 ⓔ $4y+17$

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

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

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

5. ${\,\mathrm{7x}}^{2}+5{x}^{2}$

1. $-12w+18w$

2. $\mathrm{4a}-9a$

3. $28x-(-12x)$

4. $-5b-17b$

5. $12a+5b-22a$

1. $2{a}^{2}+{b}^{2}-6{a}^{2}$

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

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

4. $\mathrm{12a}+8b$

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

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

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

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

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

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

1. $(4{m}^{2}-6m-3)-(2{m}^{2}+m-7)$

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

3. $(12{s}^{2}-15s)-(s-9)$

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

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

1. Find the sum of $(2{p}^{3}-8)$ and $({p}^{2}+9p+18)$.

2. Find the sum of $(8{a}^{3}-8a)$ and $({a}^{2}+6a+12)$.

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

4. Find the difference of$({c}^{2}+4c-33)$ and$({c}^{2}-8c+12)$.

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

1. $(7{m}^{2}+mn-8{n}^{2})+(3{m}^{2}+2mn)$

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

3. $({u}^{2}-{v}^{2})-({u}^{2}-4uv-3{v}^{2})$

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

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

1. Evaluate $8{y}^{2}-3y+2$ when:

ⓐ $y=5$ ⓑ $y=-2$ ⓒ $y=0$

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

ⓐ $y=-4$ ⓑ $y=1$ ⓒ $y=0$

3. Evaluate $4-36x$ when:

ⓐ $x=3$ ⓑ $x=0$ ⓒ $x=-1$

4. 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.

5. 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.

1. Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

2. Jonathan thinks that $\frac{1}{3}$ and $\frac{1}{x}$ are both monomials. What is wrong with his reasoning?