adding and subtracting polynomials practice and problem solving c

Adding and Subtracting Polynomials

While addition and subtraction of polynomials, we simply add or subtract the terms of the same power. The power of variables in a polynomial is always a whole number, power can not be negative, irrational, or a fraction. It is straightforward to add or subtract two polynomials. A  polynomial  is a mathematics expression written in the form of \(a_0x^n + a_1x^{n-1} + a_2x^{n-2} + ...... + a_nx^{0}\).

The above expression is also called polynomial in  standard form , where \(a_0, a_1, a_2.........a_n\) are constants, and n is a  whole number . For example x 2  + 2x + 3,   5x 4  - 4x 2  + 3x +1 and 7x - √3 are polynomials.

How Can We Add Polynomials?

The addition of polynomials is simple. While adding polynomials, we simply add like terms. We can use columns to match the correct terms together in a complicated sum. Keep two rules in mind while performing the addition of polynomials.

• Rule 1:   Always take like terms together while performing addition .
• Rule 2:   Signs of all the polynomials remain the same.

For example, Add 2x 2 + 3x +2 and 3x 2 - 5x -1

• Step 1: Arranging the polynomial in standard form. In this case, they are already in their standard forms.
• Step 2:  Like terms in the above two polynomials are: 2x 2  and 3x 2 ; 3x and -5x; 2 and -1.
• Step 3: Calculations with signs remaining same:

Like Terms are terms whose variables, along with their exponents, are the same. For example, 2x, 7x, -2x, etc are all like variables.

Unlike Terms

Unlike Terms are terms whose either variables, exponents, or both variables and exponents are the not same. For example, 2, 7x 2 , -2y 2 , etc are all unlike variables.

Subtraction of Polynomials

The subtraction of polynomials is as simple as the addition of polynomials. Using columns would help us to match the correct terms together in a complicated subtraction . While subtracting polynomials, separate the like terms and simply subtract them. Keep two rules in mind while performing the subtraction of polynomials.

• Rule 1:  Always take like terms together while performing subtraction.
• Rule 2:  Signs of all the terms of the subtracting polynomial will change, + changes to - and - changes to +.

For example, we have to subtract 2x 2 + 3x +2 from 3x 2 - 5x -1

• Step 2:  Like terms in the above two polynomials are: 2x 2  and 3x 2 ;3x and -5x;2 and  -1
• Step 3:  Enclose the part of the polynomial which to be deducted in parentheses with a negative (-) sign prefixed. Then, remove the parentheses by changing the sign of each term of the polynomial expression.
• Step 4: Calculations after altering the signs of the subtracting polynomials:

Steps for Adding and Subtracting Polynomials

The addition or subtraction of polynomials is very simple to perform, all we need to do is to keep some steps in mind. To perform the addition and subtraction operation on the polynomials, the polynomials can be arranged vertically for complex expressions. For simpler calculations, we can perform the operation using the horizontal arrangement.

Adding and Subtracting Polynomials Horizontally

Polynomials can be added and subtracted in horizontal arrangement using the steps given below,

• Step 1: Arrange the polynomials in their standard form.
• Step 2: Place the polynomial next to each other horizontally. 
• Step 3: First separate the like terms. 
• Step 4:  Arrange the like terms together.
• Step 5:  Signs of all the polynomials remain the same in addition. While in Subtraction, the signs of the terms in subtracting polynomial change.
• Step 6:  Perform the calculations.

Adding and Subtracting Polynomials Vertically

Polynomials can be added and subtracted in vertical arrangement using the steps given below,

• Step 1: Arrange the polynomials in their standard form
• Step 2: Place the polynomials in a vertical arrangement, with the like terms placed one above the other in both the polynomials.
• Step 3: We can represent the missing power term in the standard form with "0" as the coefficient to avoid confusion while arranging terms.
• Step 4:  Signs of all the polynomials remain the same in addition. While in Subtraction, the signs of the terms in subtracting polynomial change.
• Step 5:  Perform the calculations

By following these steps we can solve adding and subtracting polynomials.

Example: (3x 3 + x 2 - 2x -1) + (x 3  + 6x + 3).

The given polynomials are arranged in their standard forms.

Addition performed horizontally:

• Step 1:  Separate the like terms: 3x 3  and x 3 ; x 2 ; -2x and 6x; -1 and 3
• Step 2:  Arrange the like terms together: 3x 3  + x 3 + x 2  + (-2x + 6x) + (-1 + 3)
• Step 3: Perform the calculations: (3 + 1)x 3  + x 2  + (-2 + 6)x + (-1 + 3)= 4x 3  + x 2  + 4x + 2

Addition performed vertically:

• Step 1:  Arrange both the polynomials one above the other with like terms place one above the other. We can represent the missing power term in the standard form with "0" as the coefficient to avoid confusion while arranging terms.
• Step 2: Perform the calculations.

\[ \begin{align} \ \ 3x^3 + x^2 - 2x -1 \\ + \ x^3 + 0x^2 + 6x + 3 \\ \hline \\ 4x^3 + x^2 + 4x + 2 \\ \hline \end{align}\]

• The highest power of the variable in a polynomial is called the degree of the polynomial. 
• The algebraic expressions having negative or irrational power of the variable are not polynomials.
• Addition and subtraction in polynomials can only be performable on like terms. 

Challenging Question on Adding and Subtracting Polynomials

Solved Examples

Example 1: Add two polynomials, 3x + 2y, and 4y + 5z to find the solution. 

Given polynomials are (3x + 2y) + (4y + 5z). Here like terms are only 2y and 4y. So addition can only be performed on these two terms, the other terms 3x and 5z will not get affected. 3x + (2y + 4y) + 5z = 3x + (2 + 4)y + 5z = 3x + 6y + 5z

Therefore, answer is 3x + 6y + 5z.

Example 2: Add the polynomials 3x 2 + 4y 2  - 2z 2  + 1 and -x 2  -7y 2  + 3, and subtract the result from 5x 2  + y 2 - 8z 2  - 6, to find if the sum of coefficients of all the variables is 9.

Let us add the first two polynomials.

(3x 2 + 4y 2  - 2z 2  + 1) + (-x 2  -7y 2  + 3)

= (3 - 1)x 2  + (4 - 7)y 2  - 2z 2  + (1 + 3)

= 2x 2  - 3y 2  -2z 2  + 4

Subtract the above polynomial

5x 2  + y 2  - 8z 2 - 6 - (2x 2  - 3y 2  -2z 2  + 4)

= 5x 2  + y 2  - 8z 2 - 6 - 2x 2  + 3y 2  + 2z 2  - 4

= (5 - 2)x 2  + (1 + 3)y 2  + (-8 + 2)z 2  - 10

= 3x 2  + 4y 2  - 6z 2  - 10

Sum of the coefficients of all the variables is 3 + 4 - 6 = 1. Therefore, No, the sum of all coefficients of variables is not 9.

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FAQs on Adding and Subtracting Polynomials

How do we add or subtract polynomials.

Adding or subtracting polynomials is simple. While adding or subtracting polynomials we need to keep the rules for adding and subtracting a polynomial in mind. The rules can be explained as,

• Rule 1:   Always take like terms together while performing addition or subtraction.
• Rule 2: Signs of all the polynomials remain the same in addition. While in Subtraction, the signs of the subtracting polynomials change.

What are Binomials?

Binomials are polynomials that contain only two terms. For example x 2  + y 2  and 3x + 2y are binomials. For example, x + y +  z is not a binomial.

What is the Main Thing to Remember When you are Adding and Subtracting Polynomials?

The main thing to remember while performing addition and subtraction on polynomials is:

• to keep in mind the concept of like terms
• when a polynomial multiplied with a negative sign, all the signs will be changed. i.e., + to - and - to +

How do you Combine Like Terms?

While  combining like terms , such as 2x and 7x, we simply add their coefficients. For example, 2x + 7x = (2+7)x = 9x.

What are Like Terms?

Can you combine terms with different exponents.

No, you can only combine terms with the exact same variable and the exact same exponent. That means you can only combine squared variable terms with squared variable terms, cubed variable terms with cubed variable terms, etc.

Adding and Subtracting Polynomials

A polynomial looks like this:

To add polynomials we simply add any like terms together ... so what is a like term?

Like Terms are terms whose variables (and their exponents such as the 2 in x 2 ) are the same.

In other words, terms that are "like" each other.

Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be different.

are all like terms because the variables are all x

are all like terms because the variables are all xy 2

Example: These are NOT like terms because the variables and/or their exponents are different:

Adding polynomials.

• Place like terms together
• Add the like terms

Example: Add   2x 2 + 6x + 5   and   3x 2 - 2x - 1

Here is an animated example:

(Note: there was no "like term" for the -7 in the other polynomial, so we didn't have to add anything to it. )

Adding in Columns

We can also add them in columns like this:

Adding Several Polynomials

We can add several polynomials together like that.

Example: Add    (2x 2 + 6y + 3xy)  ,   (3x 2 - 5xy - x)   and   (6xy + 5)

Line them up in columns and add:

Using columns helps us to match the correct terms together in a complicated sum.

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.

Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the "xy" term any more.

6.1 Add and Subtract Polynomials

Learning objectives.

By the end of this section, you will be able to:

• 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

Be Prepared 6.1

Before you get started, take this readiness quiz.

Simplify: 8 x + 3 x . 8 x + 3 x . If you missed this problem, review Example 1.24 .

Be Prepared 6.2

Subtract: ( 5 n + 8 ) − ( 2 n − 1 ) . ( 5 n + 8 ) − ( 2 n − 1 ) . If you missed this problem, review Example 1.139 .

Be Prepared 6.3

Write in expanded form: a 5 . a 5 . If you missed this problem, review Example 1.14 .

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. When it is of the form a x m a x m , where a a is a constant and m m is a whole number, it is called a monomial. Some examples of monomial are 8 , −2 x 2 , 4 y 3 , and 11 z 7 8 , −2 x 2 , 4 y 3 , and 11 z 7 .

A monomial is a term of the form a x m a x m , where a a is a constant and m m is a positive whole number.

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.

Polynomials

polynomial —A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

• monomial —A polynomial with exactly one term is called a monomial.
• binomial —A polynomial with exactly two terms is called a binomial.
• trinomial —A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomial b + 1 4 y 2 − 7 y + 2 4 x 4 + x 3 + 8 x 2 − 9 x + 1 Monomial 14 8 y 2 −9 x 3 y 5 −13 Binomial a + 7 4 b − 5 y 2 − 16 3 x 3 − 9 x 2 Trinomial x 2 − 7 x + 12 9 y 2 + 2 y − 8 6 m 4 − m 3 + 8 m z 4 + 3 z 2 − 1 Polynomial b + 1 4 y 2 − 7 y + 2 4 x 4 + x 3 + 8 x 2 − 9 x + 1 Monomial 14 8 y 2 −9 x 3 y 5 −13 Binomial a + 7 4 b − 5 y 2 − 16 3 x 3 − 9 x 2 Trinomial x 2 − 7 x + 12 9 y 2 + 2 y − 8 6 m 4 − m 3 + 8 m 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 .

Example 6.1

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

• ⓐ 4 y 2 − 8 y − 6 4 y 2 − 8 y − 6
• ⓑ −5 a 4 b 2 −5 a 4 b 2
• ⓒ 2 x 5 − 5 x 3 − 9 x 2 + 3 x + 4 2 x 5 − 5 x 3 − 9 x 2 + 3 x + 4
• ⓓ 13 − 5 m 3 13 − 5 m 3

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

ⓐ 5 b 5 b ⓑ 8 y 3 − 7 y 2 − y − 3 8 y 3 − 7 y 2 − y − 3 ⓒ −3 x 2 − 5 x + 9 −3 x 2 − 5 x + 9 ⓓ 81 − 4 a 2 81 − 4 a 2 ⓔ −5 x 6 −5 x 6

ⓐ 27 z 3 − 8 27 z 3 − 8 ⓑ 12 m 3 − 5 m 2 − 2 m 12 m 3 − 5 m 2 − 2 m ⓒ 5 6 5 6 ⓓ 8 x 4 − 7 x 2 − 6 x − 5 8 x 4 − 7 x 2 − 6 x − 5 ⓔ − n 4 − n 4

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.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

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

Find the degree of the following polynomials.

• ⓐ 10 y 10 y
• ⓑ 4 x 3 − 7 x + 5 4 x 3 − 7 x + 5
• ⓓ −8 b 2 + 9 b − 2 −8 b 2 + 9 b − 2
• ⓔ 8 x y 2 + 2 y 8 x y 2 + 2 y

Find the degree of the following polynomials:

ⓐ −15 b −15 b ⓑ 10 z 4 + 4 z 2 − 5 10 z 4 + 4 z 2 − 5 ⓒ 12 c 5 d 4 + 9 c 3 d 9 − 7 12 c 5 d 4 + 9 c 3 d 9 − 7 ⓓ 3 x 2 y − 4 x 3 x 2 y − 4 x ⓔ −9 −9

ⓐ 52 52 ⓑ a 4 b − 17 a 4 a 4 b − 17 a 4 ⓒ 5 x + 6 y + 2 z 5 x + 6 y + 2 z ⓓ 3 x 2 − 5 x + 7 3 x 2 − 5 x + 7 ⓔ − a 3 − a 3

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

Add: 25 y 2 + 15 y 2 25 y 2 + 15 y 2 .

Add: 12 q 2 + 9 q 2 . 12 q 2 + 9 q 2 .

Add: −15 c 2 + 8 c 2 . −15 c 2 + 8 c 2 .

Example 6.4

Subtract: 16 p − ( −7 p ) 16 p − ( −7 p ) .

Subtract: 8 m − ( −5 m ) . 8 m − ( −5 m ) .

Subtract: −15 z 3 − ( −5 z 3 ) . −15 z 3 − ( −5 z 3 ) .

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

Example 6.5

Simplify: c 2 + 7 d 2 − 6 c 2 c 2 + 7 d 2 − 6 c 2 .

Add: 8 y 2 + 3 z 2 − 3 y 2 . 8 y 2 + 3 z 2 − 3 y 2 .

Try It 6.10

Add: 3 m 2 + n 2 − 7 m 2 . 3 m 2 + n 2 − 7 m 2 .

Example 6.6

Simplify: u 2 v + 5 u 2 − 3 v 2 u 2 v + 5 u 2 − 3 v 2 .

Try It 6.11

Simplify: m 2 n 2 − 8 m 2 + 4 n 2 . m 2 n 2 − 8 m 2 + 4 n 2 .

Try It 6.12

Simplify: p q 2 − 6 p − 5 q 2 . p q 2 − 6 p − 5 q 2 .

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

Find the sum: ( 5 y 2 − 3 y + 15 ) + ( 3 y 2 − 4 y − 11 ) . ( 5 y 2 − 3 y + 15 ) + ( 3 y 2 − 4 y − 11 ) .

Try It 6.13

Find the sum: ( 7 x 2 − 4 x + 5 ) + ( x 2 − 7 x + 3 ) . ( 7 x 2 − 4 x + 5 ) + ( x 2 − 7 x + 3 ) .

Try It 6.14

Find the sum: ( 14 y 2 + 6 y − 4 ) + ( 3 y 2 + 8 y + 5 ) . ( 14 y 2 + 6 y − 4 ) + ( 3 y 2 + 8 y + 5 ) .

Example 6.8

Find the difference: ( 9 w 2 − 7 w + 5 ) − ( 2 w 2 − 4 ) . ( 9 w 2 − 7 w + 5 ) − ( 2 w 2 − 4 ) .

Try It 6.15

Find the difference: ( 8 x 2 + 3 x − 19 ) − ( 7 x 2 − 14 ) . ( 8 x 2 + 3 x − 19 ) − ( 7 x 2 − 14 ) .

Try It 6.16

Find the difference: ( 9 b 2 − 5 b − 4 ) − ( 3 b 2 − 5 b − 7 ) . ( 9 b 2 − 5 b − 4 ) − ( 3 b 2 − 5 b − 7 ) .

Example 6.9

Subtract: ( c 2 − 4 c + 7 ) ( c 2 − 4 c + 7 ) from ( 7 c 2 − 5 c + 3 ) ( 7 c 2 − 5 c + 3 ) .

Try It 6.17

Subtract: ( 5 z 2 − 6 z − 2 ) ( 5 z 2 − 6 z − 2 ) from ( 7 z 2 + 6 z − 4 ) ( 7 z 2 + 6 z − 4 ) .

Try It 6.18

Subtract: ( x 2 − 5 x − 8 ) ( x 2 − 5 x − 8 ) from ( 6 x 2 + 9 x − 1 ) ( 6 x 2 + 9 x − 1 ) .

Example 6.10

Find the sum: ( u 2 − 6 u v + 5 v 2 ) + ( 3 u 2 + 2 u v ) ( u 2 − 6 u v + 5 v 2 ) + ( 3 u 2 + 2 u v ) .

Try It 6.19

Find the sum: ( 3 x 2 − 4 x y + 5 y 2 ) + ( 2 x 2 − x y ) ( 3 x 2 − 4 x y + 5 y 2 ) + ( 2 x 2 − x y ) .

Try It 6.20

Find the sum: ( 2 x 2 − 3 x y − 2 y 2 ) + ( 5 x 2 − 3 x y ) ( 2 x 2 − 3 x y − 2 y 2 ) + ( 5 x 2 − 3 x y ) .

Example 6.11

Find the difference: ( p 2 + q 2 ) − ( p 2 + 10 p q − 2 q 2 ) ( p 2 + q 2 ) − ( p 2 + 10 p q − 2 q 2 ) .

Try It 6.21

Find the difference: ( a 2 + b 2 ) − ( a 2 + 5 a b − 6 b 2 ) ( a 2 + b 2 ) − ( a 2 + 5 a b − 6 b 2 ) .

Try It 6.22

Find the difference: ( m 2 + n 2 ) − ( m 2 − 7 m n − 3 n 2 ) ( m 2 + n 2 ) − ( m 2 − 7 m n − 3 n 2 ) .

Example 6.12

Simplify: ( a 3 − a 2 b ) − ( a b 2 + b 3 ) + ( a 2 b + a b 2 ) ( a 3 − a 2 b ) − ( a b 2 + b 3 ) + ( a 2 b + a b 2 ) .

Try It 6.23

Simplify: ( x 3 − x 2 y ) − ( x y 2 + y 3 ) + ( x 2 y + x y 2 ) ( x 3 − x 2 y ) − ( x y 2 + y 3 ) + ( x 2 y + x y 2 ) .

Try It 6.24

Simplify: ( p 3 − p 2 q ) + ( p q 2 + q 3 ) − ( p 2 q + p q 2 ) ( p 3 − p 2 q ) + ( p q 2 + q 3 ) − ( p 2 q + p q 2 ) .

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

Evaluate 5 x 2 − 8 x + 4 5 x 2 − 8 x + 4 when

• ⓐ x = 4 x = 4
• ⓑ x = −2 x = −2
• ⓒ x = 0 x = 0

Try It 6.25

Evaluate: 3 x 2 + 2 x − 15 3 x 2 + 2 x − 15 when

• ⓐ x = 3 x = 3
• ⓑ x = −5 x = −5

Try It 6.26

Evaluate: 5 z 2 − z − 4 5 z 2 − z − 4 when

• ⓐ z = −2 z = −2
• ⓑ z = 0 z = 0
• ⓒ z = 2 z = 2

Example 6.14

The polynomial −16 t 2 + 250 −16 t 2 + 250 gives the height of a ball t t seconds after it is dropped from a 250 foot tall building. Find the height after t = 2 t = 2 seconds.

Try It 6.27

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

Try It 6.28

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

Example 6.15

The polynomial 6 x 2 + 15 x y 6 x 2 + 15 x y 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 = 4 x = 4 feet and y = 6 y = 6 feet.

Try It 6.29

The polynomial 6 x 2 + 15 x y 6 x 2 + 15 x y 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 x = 6 feet and y = 4 y = 4 feet.

Try It 6.30

The polynomial 6 x 2 + 15 x y 6 x 2 + 15 x y 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 x = 5 feet and y = 8 y = 8 feet.

Access these online resources for additional instruction and practice with adding and subtracting polynomials.

• Add and Subtract Polynomials 1
• Add and Subtract Polynomials 2
• Add and Subtract Polynomial 3
• Add and Subtract Polynomial 4

Section 6.1 Exercises

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 81 b 5 − 24 b 3 + 1 ⓑ 5 c 3 + 11 c 2 − c − 8 5 c 3 + 11 c 2 − c − 8 ⓒ 14 15 y + 1 7 14 15 y + 1 7 ⓓ 5 ⓔ 4 y + 17 4 y + 17

ⓐ x 2 − y 2 x 2 − y 2 ⓑ −13 c 4 −13 c 4 ⓒ x 2 + 5 x − 7 x 2 + 5 x − 7 ⓓ x 2 y 2 − 2 x y + 8 x 2 y 2 − 2 x y + 8 ⓔ 19

ⓐ 8 − 3 x 8 − 3 x ⓑ z 2 − 5 z − 6 z 2 − 5 z − 6 ⓒ y 3 − 8 y 2 + 2 y − 16 y 3 − 8 y 2 + 2 y − 16 ⓓ 81 b 5 − 24 b 3 + 1 81 b 5 − 24 b 3 + 1 ⓔ −18 −18

ⓐ 11 y 2 11 y 2 ⓑ −73 −73 ⓒ 6 x 2 − 3 x y + 4 x − 2 y + y 2 6 x 2 − 3 x y + 4 x − 2 y + y 2 ⓓ 4 y + 17 4 y + 17 ⓔ 5 c 3 + 11 c 2 − c − 8 5 c 3 + 11 c 2 − c − 8

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

ⓐ 6 a 2 + 12 a + 14 6 a 2 + 12 a + 14 ⓑ 18 x y 2 z 18 x y 2 z ⓒ 5 x + 2 5 x + 2 ⓓ y 3 − 8 y 2 + 2 y − 16 y 3 − 8 y 2 + 2 y − 16 ⓔ −24 −24

ⓐ 9 y 3 − 10 y 2 + 2 y − 6 9 y 3 − 10 y 2 + 2 y − 6 ⓑ −12 p 4 −12 p 4 ⓒ a 2 + 9 a + 18 a 2 + 9 a + 18 ⓓ 20 x 2 y 2 − 10 a 2 b 2 + 30 20 x 2 y 2 − 10 a 2 b 2 + 30 ⓔ 17

ⓐ 14 − 29 x 14 − 29 x ⓑ z 2 − 5 z − 6 z 2 − 5 z − 6 ⓒ y 3 − 8 y 2 + 2 y − 16 y 3 − 8 y 2 + 2 y − 16 ⓓ 23 a b 2 − 14 23 a b 2 − 14 ⓔ −3 −3

ⓐ 62 y 2 62 y 2 ⓑ 15 ⓒ 6 x 2 − 3 x y + 4 x − 2 y + y 2 6 x 2 − 3 x y + 4 x − 2 y + y 2 ⓓ 10 − 9 x 10 − 9 x ⓔ m 4 + 4 m 3 + 6 m 2 + 4 m + 1 m 4 + 4 m 3 + 6 m 2 + 4 m + 1

In the following exercises, add or subtract the monomials.

7x 2 + 5 x 2 7x 2 + 5 x 2

4y 3 + 6 y 3 4y 3 + 6 y 3

−12 w + 18 w −12 w + 18 w

−3 m + 9 m −3 m + 9 m

4a − 9 a 4a − 9 a

− y − 5 y − y − 5 y

28 x − ( −12 x ) 28 x − ( −12 x )

13 z − ( −4 z ) 13 z − ( −4 z )

−5 b − 17 b −5 b − 17 b

−10 x − 35 x −10 x − 35 x

12 a + 5 b − 22 a 12 a + 5 b − 22 a

14x − 3 y − 13 x 14x − 3 y − 13 x

2 a 2 + b 2 − 6 a 2 2 a 2 + b 2 − 6 a 2

5 u 2 + 4 v 2 − 6 u 2 5 u 2 + 4 v 2 − 6 u 2

x y 2 − 5 x − 5 y 2 x y 2 − 5 x − 5 y 2

p q 2 − 4 p − 3 q 2 p q 2 − 4 p − 3 q 2

a 2 b − 4 a − 5 a b 2 a 2 b − 4 a − 5 a b 2

x 2 y − 3 x + 7 x y 2 x 2 y − 3 x + 7 x y 2

12a + 8 b 12a + 8 b

19y + 5 z 19y + 5 z

Add: 4 a , −3 b , −8 a 4 a , −3 b , −8 a

Add: 4x , 3 y , −3 x 4x , 3 y , −3 x

Subtract 5 x 6 from − 12 x 6 5 x 6 from − 12 x 6 .

Subtract 2 p 4 from − 7 p 4 2 p 4 from − 7 p 4 .

In the following exercises, add or subtract the polynomials.

( 5 y 2 + 12 y + 4 ) + ( 6 y 2 − 8 y + 7 ) ( 5 y 2 + 12 y + 4 ) + ( 6 y 2 − 8 y + 7 )

( 4 y 2 + 10 y + 3 ) + ( 8 y 2 − 6 y + 5 ) ( 4 y 2 + 10 y + 3 ) + ( 8 y 2 − 6 y + 5 )

( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x − 9 ) ( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x − 9 )

( y 2 + 9 y + 4 ) + ( −2 y 2 − 5 y − 1 ) ( y 2 + 9 y + 4 ) + ( −2 y 2 − 5 y − 1 )

( 8 x 2 − 5 x + 2 ) + ( 3 x 2 + 3 ) ( 8 x 2 − 5 x + 2 ) + ( 3 x 2 + 3 )

( 7 x 2 − 9 x + 2 ) + ( 6 x 2 − 4 ) ( 7 x 2 − 9 x + 2 ) + ( 6 x 2 − 4 )

( 5 a 2 + 8 ) + ( a 2 − 4 a − 9 ) ( 5 a 2 + 8 ) + ( a 2 − 4 a − 9 )

( p 2 − 6 p − 18 ) + ( 2 p 2 + 11 ) ( p 2 − 6 p − 18 ) + ( 2 p 2 + 11 )

( 4 m 2 − 6 m − 3 ) − ( 2 m 2 + m − 7 ) ( 4 m 2 − 6 m − 3 ) − ( 2 m 2 + m − 7 )

( 3 b 2 − 4 b + 1 ) − ( 5 b 2 − b − 2 ) ( 3 b 2 − 4 b + 1 ) − ( 5 b 2 − b − 2 )

( a 2 + 8 a + 5 ) − ( a 2 − 3 a + 2 ) ( a 2 + 8 a + 5 ) − ( a 2 − 3 a + 2 )

( b 2 − 7 b + 5 ) − ( b 2 − 2 b + 9 ) ( b 2 − 7 b + 5 ) − ( b 2 − 2 b + 9 )

( 12 s 2 − 15 s ) − ( s − 9 ) ( 12 s 2 − 15 s ) − ( s − 9 )

( 10 r 2 − 20 r ) − ( r − 8 ) ( 10 r 2 − 20 r ) − ( r − 8 )

Subtract ( 9 x 2 + 2 ) ( 9 x 2 + 2 ) from ( 12 x 2 − x + 6 ) ( 12 x 2 − x + 6 ) .

Subtract ( 5 y 2 − y + 12 ) ( 5 y 2 − y + 12 ) from ( 10 y 2 − 8 y − 20 ) ( 10 y 2 − 8 y − 20 ) .

Subtract ( 7 w 2 − 4 w + 2 ) ( 7 w 2 − 4 w + 2 ) from ( 8 w 2 − w + 6 ) ( 8 w 2 − w + 6 ) .

Subtract ( 5 x 2 − x + 12 ) ( 5 x 2 − x + 12 ) from ( 9 x 2 − 6 x − 20 ) ( 9 x 2 − 6 x − 20 ) .

Find the sum of ( 2 p 3 − 8 ) ( 2 p 3 − 8 ) and ( p 2 + 9 p + 18 ) ( p 2 + 9 p + 18 ) .

Find the sum of ( q 2 + 4 q + 13 ) ( q 2 + 4 q + 13 ) and ( 7 q 3 − 3 ) ( 7 q 3 − 3 ) .

Find the sum of ( 8 a 3 − 8 a ) ( 8 a 3 − 8 a ) and ( a 2 + 6 a + 12 ) ( a 2 + 6 a + 12 ) .

Find the sum of ( b 2 + 5 b + 13 ) ( b 2 + 5 b + 13 ) and ( 4 b 3 − 6 ) ( 4 b 3 − 6 ) .

Find the difference of ( w 2 + w − 42 ) ( w 2 + w − 42 ) and ( w 2 − 10 w + 24 ) ( w 2 − 10 w + 24 ) .

Find the difference of ( z 2 − 3 z − 18 ) ( z 2 − 3 z − 18 ) and ( z 2 + 5 z − 20 ) ( z 2 + 5 z − 20 ) .

Find the difference of ( c 2 + 4 c − 33 ) ( c 2 + 4 c − 33 ) and ( c 2 − 8 c + 12 ) ( c 2 − 8 c + 12 ) .

Find the difference of ( t 2 − 5 t − 15 ) ( t 2 − 5 t − 15 ) and ( t 2 + 4 t − 17 ) ( t 2 + 4 t − 17 ) .

( 7 x 2 − 2 x y + 6 y 2 ) + ( 3 x 2 − 5 x y ) ( 7 x 2 − 2 x y + 6 y 2 ) + ( 3 x 2 − 5 x y )

( −5 x 2 − 4 x y − 3 y 2 ) + ( 2 x 2 − 7 x y ) ( −5 x 2 − 4 x y − 3 y 2 ) + ( 2 x 2 − 7 x y )

( 7 m 2 + m n − 8 n 2 ) + ( 3 m 2 + 2 m n ) ( 7 m 2 + m n − 8 n 2 ) + ( 3 m 2 + 2 m n )

( 2 r 2 − 3 r s − 2 s 2 ) + ( 5 r 2 − 3 r s ) ( 2 r 2 − 3 r s − 2 s 2 ) + ( 5 r 2 − 3 r s )

( a 2 − b 2 ) − ( a 2 + 3 a b − 4 b 2 ) ( a 2 − b 2 ) − ( a 2 + 3 a b − 4 b 2 )

( m 2 + 2 n 2 ) − ( m 2 − 8 m n − n 2 ) ( m 2 + 2 n 2 ) − ( m 2 − 8 m n − n 2 )

( u 2 − v 2 ) − ( u 2 − 4 u v − 3 v 2 ) ( u 2 − v 2 ) − ( u 2 − 4 u v − 3 v 2 )

( j 2 − k 2 ) − ( j 2 − 8 j k − 5 k 2 ) ( j 2 − k 2 ) − ( j 2 − 8 j k − 5 k 2 )

( p 3 − 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) − ( 3 p 2 q + p q 2 ) ( p 3 − 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) − ( 3 p 2 q + p q 2 )

( a 3 − 2 a 2 b ) + ( a b 2 + b 3 ) − ( 3 a 2 b + 4 a b 2 ) ( a 3 − 2 a 2 b ) + ( a b 2 + b 3 ) − ( 3 a 2 b + 4 a b 2 )

( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 ) ( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 )

( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 ) ( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 )

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

Evaluate 8 y 2 − 3 y + 2 8 y 2 − 3 y + 2 when:

ⓐ y = 5 y = 5 ⓑ y = −2 y = −2 ⓒ y = 0 y = 0

Evaluate 5 y 2 − y − 7 5 y 2 − y − 7 when:

ⓐ y = −4 y = −4 ⓑ y = 1 y = 1 ⓒ y = 0 y = 0

Evaluate 4 − 36 x 4 − 36 x when:

ⓐ x = 3 x = 3 ⓑ x = 0 x = 0 ⓒ x = −1 x = −1

Evaluate 16 − 36 x 2 16 − 36 x 2 when:

ⓐ x = −1 x = −1 ⓑ x = 0 x = 0 ⓒ x = 2 x = 2

A painter drops a brush from a platform 75 feet high. The polynomial −16 t 2 + 75 −16 t 2 + 75 gives the height of the brush t t seconds after it was dropped. Find the height after t = 2 t = 2 seconds.

A girl drops a ball off a cliff into the ocean. The polynomial −16 t 2 + 250 −16 t 2 + 250 gives the height of a ball t t seconds after it is dropped from a 250-foot tall cliff. Find the height after t = 2 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 + 420 p . −4 p 2 + 420 p . Find the revenue received when p = 60 p = 60 dollars.

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 + 420 p . −4 p 2 + 420 p . Find the revenue received when p = 90 p = 90 dollars.

Everyday Math

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of x x miles per hour is given by the polynomial − 1 150 x 2 + 1 3 x − 1 150 x 2 + 1 3 x . Find the fuel efficiency when x = 30 mph x = 30 mph .

Stopping Distance The number of feet it takes for a car traveling at x x miles per hour to stop on dry, level concrete is given by the polynomial 0.06 x 2 + 1.1 x 0.06 x 2 + 1.1 x . Find the stopping distance when x = 40 mph x = 40 mph .

Rental Cost The cost to rent a rug cleaner for d d days is given by the polynomial 5.50 d + 25 5.50 d + 25 . Find the cost to rent the cleaner for 6 days.

Height of Projectile The height (in feet) of an object projected upward is given by the polynomial −16 t 2 + 60 t + 90 −16 t 2 + 60 t + 90 where t t represents time in seconds. Find the height after t = 2.5 t = 2.5 seconds.

Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial 9 5 c + 32 9 5 c + 32 where c c represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when c = 65 ° . c = 65 ° .

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 6 y 2 + 5 y 4 is 11 y 6 11 y 6 . What is wrong with her reasoning?

Jonathan thinks that 1 3 1 3 and 1 x 1 x are both monomials. What is wrong with his reasoning?

ⓐ 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.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction

• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Elementary Algebra 2e
• Publication date: Apr 22, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/elementary-algebra-2e/pages/6-1-add-and-subtract-polynomials

© Jul 24, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Polynomial Worksheets

These worksheets focus on the topics typically covered in Algebra I

• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet

Algebra 2 Polynomial Worksheets

Ultimate math solver (free) free algebra solver ... type anything in there, popular pages @ mathwarehouse.com.

Subtraction of Polynomials – Example and Practice Problems

Polynomial subtraction is one of the easiest mathematical operations that we can perform with polynomials. To perform subtraction of polynomials, we have to identify like terms and subtract the coefficients of the like terms.

Here, we will look at a summary of the subtraction of polynomials and the process used. In addition, we will look at several examples with answers of subtraction of polynomials in order to fully master this topic.

Relevant for …

Exploring examples of subtraction of polynomials.

See examples

Summary of subtraction of polynomials

Subtraction of polynomials – examples with answers, subtraction of polynomials – practice problems.

To subtract two or more polynomials, we just have to combine like terms and consider the order of operations. Something important that must be taken into account is to distinguish the terms with “plus” and “minus” signs in each polynomial.

We follow the following steps to subtract polynomials:

Step 1:  Remove all parentheses. To facilitate visualization, it is advisable to write the problem and each process vertically. When we remove the parentheses, we have to distribute the negative sign, which will cause each of the terms to change the sign.

Step 2:  Combine like terms. If we write the steps vertically, combining like terms is easier. Remember that like terms are terms that have the same variables with the same exponents.

The following examples have their respective solutions, so you can study them carefully and master the process of simplifying algebraic expressions.

Solve the subtraction of polynomials: $latex (6x+8y)-(3x-2y)$.

We have to remove the parentheses. To do this, we have to take into account the negative sign in front of the second polynomial, so we change the sign of all the terms of the second polynomial. Then, we have to group like terms:

$latex (6x+8y)-(3x-2y)$

$latex =6x+8y-3x+2y$

$latex =6x-3x+8y+2y$

$latex =3x+10y$

These terms are no longer similar since they do not have the same variable, so we cannot combine them.

Solve the subtraction $latex (6x+8y)-(3x-2y)$ vertically.

We can subtract polynomials vertically by placing each variable in its own column. Therefore, we use the first column for  x  and the second for  y.  Since we are subtracting the second polynomial, we have to change the sign of all its terms:

$latex 6x+8y$

$latex -3x+2y$

___________

$latex 3x+10y$

Clearly, we got the same answer as when we subtracted horizontally. It is possible to solve the subtraction of polynomials using any format. Just choose the format with which you feel most comfortable.

Generally, when we have simple polynomials, the horizontal format is easier. However, for longer and more complicated polynomials, subtracting vertically can make the process easier.

Solve the subtraction of polynomials: $$(4{{x}^3}+2{{x}^2}-4x+6)-(2{{x}^3}+4{{x}^2}+6x-7)$$

We can start by performing the subtraction horizontally. Therefore, we remove the parentheses taking into account the “minus” sign and then combine like terms:

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

$$=4{{x}^3}+2{{x}^2}-4x+6-2{{x}^3}-4{{x}^2}-6x+7$$

$$=4{{x}^3}-2{{x}^3}+2{{x}^2}-4{{x}^2}-4x-6x+6+7$$

$latex =2{{x}^3}-2{{x}^2}-10x+13$

Now, we can perform the subtraction vertically. We separate each exponent by columns and change the sign of all the terms of the second polynomial since we have the “minus” sign in front:

$latex 4{{x}^3}+2{{x}^2}-4x+6$

$latex -2{{x}^3}-4{{x}^2}-6x+7$

___________________

$latex 2{{x}^3}-2{{x}^2}-10x+13$

Solve the subtraction $$(-5{{x}^3}+6{{x}^2}-4x)-(3{{x}^3}-5{{x}^2}-6x)$$

We are going to perform the subtraction horizontally. Therefore, we remove the parentheses by changing the signs of the terms of the second polynomial since they have the “minus” sign in front of them. Then we simplify by combining like terms:

$$(-5{{x}^3}+6{{x}^2}-4x)-(3{{x}^3}-5{{x}^2}-6x)$$

$$=-5{{x}^3}+6{{x}^2}-4x-3{{x}^3}+5{{x}^2}+6x$$

$$=-5{{x}^3}-3{{x}^3}+6{{x}^2}+5{{x}^2}-4x+6x$$

$latex =-8{{x}^3}+11{{x}^2}+2x$

Now, we perform the subtraction vertically. We have to assign a column to each exponent and we have to change the sign to the terms of the second polynomial:

$latex -5{{x}^3}+6{{x}^2}-4x$

$latex -3{{x}^3}+5{{x}^2}+6x$

________________

$latex -8{{x}^3}+11{{x}^2}+2x$

Solve the following operation: $latex (4{{x}^2}+2x-5)-(5{{x}^2}-4x-4)$ $latex -(-6{{x}^2}+3x+10)$.

In this case, it will be easier if we solve the subtraction of polynomials using the vertical format. We separate the variables in different columns depending on the exponent and change the sign of all the terms of the polynomials that are being subtracted:

$latex 4{{x}^2}+2x-5$

$latex -5{{x}^2}+4x+4$

$latex 6{{x}^2}-3x-10$

$latex 5{{x}^2}+3x-11$

Practice subtraction of polynomials with the following problems. Solve the subtractions and choose your answer. Click “Check” to verify that you selected the correct answer.

Solve the subtraction $latex (4{{x}^2}+5x)-({{x}^2}-2x)$.

Choose an answer

Solve the subtraction $latex (-2x+5y)-(-6x+7y)$.

Solve the subtraction $latex (6{{x}^3}-2{{x}^2}+8x)-(4{{x}^3}-11x+10)$., solve the subtraction $latex ({{x}^3}+2{{x}^2}+10)-(-6{{x}^3}-7{{x}^2}-5)$..

Interested in learning more about operations with polynomials? Take a look at these pages:

• Examples of Addition of Polynomials
• Examples of Multiplication of Monomials
• Examples of Multiplication of Polynomials
• Examples of Division of Polynomials

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Worksheet on Adding and Subtracting Polynomials

Practice the problems given in the worksheet on adding and subtracting polynomials. The questions are based on different types of word problems on addition and subtraction of polynomials.

1. Solve the following subtraction:

(i) Take – (3/2)a + b – c from (1/2)a – (1/3)b – (3/2)c

(iii) Take 3a + 2b – 6c from 8a – 4b – 2c

2. (i) What should be added to 3m to get 5m?

(ii) What should be added to -p to get 4p?

(iii) What should be added to x - y to get 2x + y?

3.  From the sum of a + b – 2c and 2a – b + c, subtract a + b + c.

4.  Subtract the sum of p + q and p – r from the sum of p – 2r and p + q + r.

6. From the sum of a - b + 11 and –b – 9, subtract 2a - 3b - 1.

Answers for the worksheet on adding and subtracting polynomials are given below to check the exact answers of the above word problems.

1. (i) 2a – (4/3)b – (1/2)c

(iii) 5a – 6b + 4c

(iii) x + 2y

3. 2a – b – 2c

● Terms of an Algebraic Expression - Worksheet

Worksheet on Types of Algebraic Expressions

Worksheet on Degree of a Polynomial

Worksheet on Addition of Polynomials

Worksheet on Subtraction of Polynomials

Worksheet on Addition and Subtraction of Polynomials

Worksheet on Multiplying Monomials

Worksheet on Multiplying Monomial and Binomial

Worksheet on Multiplying Monomial and Polynomial

Worksheet on Multiplying Binomials

Worksheet on Dividing Monomials

6th Grade Math Practice

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