10.1 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

In Evaluate, Simplify, and Translate Expressions, you 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. A monomial, or a sum and/or difference of monomials, is called a polynomial.

Notice the roots:

• poly- means many

• mono- means one

• bi- means two

• tri- means three

Here are some examples of polynomials:

Notice that every monomial, binomial, and trinomial is also a polynomial. They are 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 10.1

Solution

1. Count the terms in each polynomial. A monomial has one term, a binomial has two, a trinomial has three, and any polynomial with more than three terms has another type.
2. Classify part ⓐ. The expression $8{x}^{2}-7x-9$ has three terms, so it is a trinomial.
3. Classify part ⓑ. The expression $-5{a}^{4}$ has one term, so it is a monomial.
4. Classify part ⓒ. The expression ${x}^{4}-7{x}^{3}-6{x}^{2}+5x+2$ has five terms, so it is another polynomial.
5. Classify part ⓓ. The expression $11-4{y}^{3}$ has two terms, so it is a binomial.
6. Classify part ⓔ. The expression $n$ has one term, so it is a monomial.

Takeaway: Classify a polynomial by its number of terms: one term is a monomial, two terms a binomial, three terms a trinomial, and more than three terms another polynomial.

Determine the Degree of Polynomials

In this section, we will work with polynomials that have only one variable in each term. 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.

Remember: Any base written without an exponent has an implied exponent of $1.$

Example 10.2

Solution

1. Identify the degree rule. The degree of a polynomial is the greatest exponent on its variable terms, while a nonzero constant has degree $0$.
2. Find the degree in part ⓐ. In $4x$, the exponent of $x$ is $1$, so the polynomial has degree $1$.
3. Find the degree in part ⓑ. The greatest exponent in $3{x}^{3}-5x+7$ is $3$, so the polynomial has degree $3$.
4. Find the degree in part ⓒ. Since $-11$ is a constant, it has degree $0$ and contains no variable term.
5. Find the degree in part ⓓ. The greatest exponent in $-6{x}^{2}+9x-3$ is $2$, so the polynomial has degree $2$.
6. Find the degree in part ⓔ. The greatest exponent in $8x+2$ is $1$, so the polynomial has degree $1$.

Takeaway: The degree of a polynomial is the largest exponent on any term, so the degrees here are 1, 3, 0, 2, and 1 in order.

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. Look back at the polynomials in Example 10.2. Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.

Add and Subtract Monomials

In The Language of Algebra, you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

Example 10.3

Solution

1. Identify the like terms. Both $17{x}^{2}$ and $6{x}^{2}$ have the same variable and exponent, so their coefficients can be added.
2. Add the coefficients. The coefficient sum is $17+6=23$, which replaces the two original coefficients.
3. Write the simplified sum. Keeping the common exponent $2$ gives $23{x}^{2}$.

Takeaway: Adding like terms means adding their coefficients, so $17{x}^{2}+6{x}^{2}=23{x}^{2}$.

Example 10.4

Solution

1. Rewrite the subtraction. Subtracting a negative is equivalent to adding its opposite, so $11n-(-8n)$ becomes $11n+8n$.
2. Combine the like terms. The terms have the same variable and exponent, so add their coefficients to get $11+8=19$.
3. Write the simplified difference. The common factor $n$ remains, and the result is $19n$.

Takeaway: Subtracting a negative changes the operation to addition, so $11n-(-8n)=11n+8n=19n$.

Example 10.5

Solution

1. Group the like terms. The two $a^2$ terms are like terms because they have the same variable and exponent, but $4b^2$ has a different variable part.
2. Combine the $a^2$ coefficients. Writing the coefficient of the first term as $1$ gives $1-7=-6$.
3. Write the simplified expression. The combined term is $-6{a}^{2}$, and the unrelated term $4{b}^{2}$ remains unchanged.

Takeaway: Only terms with the same variable part are like terms, so ${a}^{2}+4{b}^{2}-7{a}^{2}=-6{a}^{2}+4{b}^{2}$.

Add and Subtract Polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.

Example 10.6

Solution

1. Remove the parentheses. Write the sum as $4{x}^{2}-5x+1+3{x}^{2}-8x-9$, preserving every sign in the second polynomial.
2. Combine the quadratic terms. The terms $4{x}^{2}$ and $3{x}^{2}$ are like terms, so $(4+3){x}^{2}=7{x}^{2}$.
3. Combine the linear terms and constants. The linear terms give $(-5-8)x=-13x$, while the constants give $1-9=-8$.
4. State the simplified sum. The figures group the $x^2$, $x$, and constant terms and end with the result $7{x}^{2}-13x-8$.

Takeaway: To add polynomials, combine coefficients within each like-term family while keeping unlike families in their original order.

Parentheses are grouping symbols. When we add polynomials as we did in Example 10.6, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

Example 10.7

Solution

1. Distribute the subtraction sign. Rewrite $(7{u}^{2}-5u+3)-(4{u}^{2}-2)$ as $7{u}^{2}-5u+3-4{u}^{2}+2$.
2. Combine the quadratic terms. Since $7{u}^{2}$ and $-4{u}^{2}$ are like terms, their coefficients give $(7-4){u}^{2}=3{u}^{2}$.
3. Combine the remaining terms. The linear term remains $-5u$, and the constants combine as $3+2=5$.
4. State the difference. The figures track the sign change and grouped like terms before displaying $3{u}^{2}-5u+5$.

Takeaway: Distribute a subtraction sign across every term of the polynomial being subtracted before combining like terms.

Example 10.8

Solution

1. Write the subtraction in order. Start with $9{m}^{2}-7m+4-({m}^{2}-3m+8)$ so it is clear which polynomial is being removed.
2. Distribute the minus sign. This gives $9{m}^{2}-7m+4-{m}^{2}+3m-8$, so the coefficients of $m^2$, $m$, and the constant all change sign.
3. Combine like terms. The quadratic coefficient is $9-1=8$, the linear coefficient is $-7+3=-4$, and the constant is $4-8=-4$.
4. State the difference. The final figures display the regrouped expression and the simplified polynomial $8{m}^{2}-4m-4$.

Takeaway: In a phrase such as “subtract A from B,” compute $B-A$ and change the sign of every term in $A$.

Evaluate a Polynomial for a Given Value

In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify.

Example 10.9

Solution

1. Substitute in part ⓐ. Replacing $x$ with $3$ gives $3(3)^{2}-9(3)+7$, and $(3)^{2}=9$.
2. Simplify part ⓐ. The expression becomes $3\cdot{}9-27+7=27-27+7=7$.
3. Substitute in part ⓑ. Replacing $x$ with $-1$ gives $3(-1)^{2}-9(-1)+7$, and $(-1)^{2}=1$.
4. Simplify part ⓑ. The expression becomes $3\cdot{}1+9+7=19$, so the two values are $7$ and $19$.

Takeaway: Substituting $x=3$ gives $7$, while substituting $x=-1$ gives $19$ because the squared term remains positive.

Example 10.10

Solution

1. Substitute the elapsed time. Replace $t$ with $3$ in the height model to obtain $-16(3)^{2}+300$.
2. Evaluate the exponent. Using the power rule for equal factors, $3^{2}=3\cdot{}3=9$, so the expression becomes $-16\cdot{}9+300$.
3. Multiply and add. The product is $-16\cdot{}9=-144$, and therefore $-144+300=156$.
4. Interpret the result. The figures show the same substitutions and arithmetic, so the object is $156$ feet above the ground after $3$ seconds.

Takeaway: When evaluating a polynomial model, substitute first and then apply the power rule before completing the arithmetic.

Section 10.1 Exercises

Section 10.1 Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the 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

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. 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.

1. ⓐ $\,z$

2. ⓑ $\,2{x}^{3}-4{x}^{2}-x-8$

3. ⓒ $\,6{x}^{2}-4x+1$

4. ⓓ $\,9-4{y}^{2}$

5. ⓔ $\,3{x}^{7}$

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

1. ⓐ $\,{y}^{3}-8$

2. ⓑ $\,9{x}^{3}-5{x}^{2}-x$

3. ⓒ $\,{x}^{4}-3{x}^{2}-4x-7$

4. ⓓ $\,-{y}^{4}$

5. ⓔ $\,w$

3. Find the degree of the following polynomials:

1. ⓐ $\,-6y$

2. ⓑ $\,4x-1$

3. ⓒ $\,3{x}^{4}+4{x}^{2}-8$

4. ⓓ $\,2{y}^{2}+3y+9$

5. ⓔ $\,-18$

1. Find the degree of the following polynomials:

1. ⓐ $\,47$

2. ⓑ $\,2{x}^{2}-8x+2$

3. ⓒ $\,{x}^{4}-16$

4. ⓓ $\,{y}^{5}-5{y}^{3}+y$

5. ⓔ $\,9{a}^{3}$

2. Add: $12{x}^{2}+5{x}^{2}.$

3. Add: $-11{y}^{2}+8{y}^{2}.$

1. Subtract: $9n-(-5n).$

2. Subtract: $-7{a}^{3}-(-5{a}^{3}).$

3. Add: $3{x}^{2}+3{y}^{2}-5{x}^{2}.$

1. Add: $2{a}^{2}+{b}^{2}-4{a}^{2}.$

2. Find the sum: $(3{x}^{2}-2x+8)+({x}^{2}-6x+2).$

3. Find the sum: $(7{y}^{2}+4y-6)+(4{y}^{2}+5y+1).$

1. Find the difference: $(6{y}^{2}+3y-1)-(3{y}^{2}-4).$

2. Find the difference: $(8{u}^{2}-7u-2)-(5{u}^{2}-6u-4).$

3. Subtract: $(4{n}^{2}-7n-3)$ from $(8{n}^{2}+5n-3).$

1. Subtract: $({a}^{2}-4a-9)$ from $(6{a}^{2}+4a-1).$

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

1. ⓐ $\,x=2$

2. ⓑ $\,x=-3$

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

1. ⓐ $\,y=-4$

2. ⓑ $\,y=0$

1. 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.

2. 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.

3. $5x+2$

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

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

3. $23{y}^{2}$

1. ${\mathrm{6x}}^{2}+\,9{x}^{2}$

2. $-12u\,+\,4u$

3. $5a\,+\,7b$

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

2. $18x-2x$

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

1. $(4{y}^{2}+10y+3)+(8{y}^{2}-6y+5)$

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

3. $(3{a}^{2}+7)+({a}^{2}-7a-18)$

1. $(6{m}^{2}-9m-3)-(2{m}^{2}+m-5)$

2. $({z}^{2}+8z+9)-({z}^{2}-3z+1)$

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

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

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

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

1. $\mathrm{Evaluate}\,8{y}^{2}-3y+2$

1. ⓐ $\,y=5$

2. ⓑ $\,y=-2$

3. ⓒ $\,y=0$

2. $\mathrm{Evaluate}\,4-36x\,\mathrm{when:}$

1. ⓐ $\,x=3$

2. ⓑ $\,x=0$

3. ⓒ $x=-1$

3. 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.

1. 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 $-\frac{1}{160}\,{x}^{2}+\frac{1}{2}\,x.$ Find the fuel efficiency when $x=40\,\mathrm{mph.}$

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