Adding and Subtracting Polynomials
Site: | Saylor Academy |
Course: | MA007: Algebra |
Book: | Adding and Subtracting Polynomials |
Printed by: | Guest user |
Date: | Tuesday, 15 July 2025, 7:45 AM |
Description

Learning Objectives
Source: OpenStax, https://openstax.org/books/elementary-algebra/pages/6-1-add-and-subtract-polynomials
This work is licensed under a Creative Commons Attribution 4.0 License.
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 \(a x^{m}\), where \(a\) is a constant and \(m\) is a whole number, it is called a monomial. Some examples of monomial are \(8,-2 x^{2}, 4 y^{3}\), and \(11 z^{7}\).
Monomials
A monomial is a term of the form \(a x^{m}\), where a is a constant and 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 | \(b+1 \) | \(4y^2−7y+2\) | \(4x^4+x^3+8x^2−9x+1\) |
---|---|---|---|
Monomial | \(14\) | \(8y^2\) | \(−9x^3y^5\) |
Binomial | \(a+7\) | \(4b−5\) | \(y^2−16\) |
Trinomial | \(x^2−7x+12\) | \(9y^2+2y−8\) | \(6m^4−m^3+8m\) |
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.
- \(4 y^{2}-8 y-6\)
- \(-5 a^{4} b^{2}\)
- \(2 x^{5}-5 x^{3}-9 x^{2}+3 x+4\)
- \(13-5 m^{3}\)
- \(q\)
Solution
Polynomial | Number of terms | Type | |
|
\(4 y^{2}-8 y-6\) | \(3\) | Trinomial |
|
\(-5 a^{4} b^{2}\) | \(1\) | Monomial |
|
\(2 x^{5}-5 x^{3}-9 x^{2}+3 x+4\) | \(5\) | Polynomial |
|
\(13-5 m^{3}\) | \(2\) | Binomial |
|
\(q\) | \(1\) | Monomial |
Try It 6.1
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
- \(5 b\)
- \(8 y^{3}-7 y^{2}-y-3\)
- \(-3 x^{2}-5 x+9\)
- \(81-4 a^{2}\)
- \(-5 x^{6}\)
Try It 6.2
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
- \(27 z^{3}-8\)
- \(12 m^{3}-5 m^{2}-2 m\)
- \(\frac{5}{6}\)
- \(8 x^{4}-7 x^{2}-6 x-5\)
- \(-n^{4}\)
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.
Monomial Degree |
\(14\) \(0\) |
\(8 y^{2}\) \(2\) |
\(-9 x^{3} y^{5}\) \(8\) |
\(-13a\) \(1\) |
Binomial Degree of each term Degree of polynomial |
\(a+7\) \(1 \qquad 0\) \(1\) |
\(4 b^{2}-5 b\) \(2 \qquad 1\) \(2\) |
\(x^{2} y^{2}-16\) \(4 \qquad 0\) \(4\) |
\(3 n^{3}-9 n^{2}\) \(3 \qquad 2\) \(3\) |
Trinomial Degree of each term Degree of polynomial |
\(x^{2}-7 x+12\) \(2 \qquad 1 \qquad 0\) \(2\) |
\(9 a^{2}+6 a b+b^{2}\) \(2 \qquad 2 \qquad 2\) \(2\) |
\(6 m^{4}-m^{3} n^{2}+8 m n^{5}\) \(4 \qquad 5 \qquad 6\) \(6\) |
\(z^{4}+3 z^{2}-1\) \(\begin{array}{lll}4 & 2 & 0\end{array}\) \(4\) |
Polynomial Degree of each term Degree of polynomial |
\(b+1\) \(1 \qquad 1 \qquad 0\) \(1\) |
\(4 y^{2}-7 y+2\) \(2 \qquad 1 \qquad 1 \qquad 0\) \(2\) |
\(4 x^{4}+x^{3}+8 x^{2}-9 x+1\) \(\begin{array}{lllll}4 & 3 & 2 & 1 & 0\end{array}\) \(4\) |
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.
- \(10 y\)
- \(4 x^{3}-7 x+5\)
- \(-15\)
- \(-8 b^{2}+9 b-2\)
- \(8 x y^{2}+2 y\)
Solution
|
\(10y\) The degree is 1. |
|
\(4 x^{3}-7 x+5\) The degree is 3. |
|
\(−15\) The degree is 0. |
|
\(-8 b^{2}+9 b-2\) The degree is 2. |
|
\(8 x y^{2}+2 y\) The degree is 3. |
Try It 6.3
Find the degree of the following polynomials:
- \(-15 b\)
- \(10 z^{4}+4 z^{2}-5\)
- \(12 c^{5} d^{4}+9 c^{3} d^{9}-7\)
- \(3 x^{2} y-4 x\)
- \(-9\)
Try It 6.4
Find the degree of the following polynomials:
- \(52\)
- \(a^{4} b-17 a^{4}\)
- \(5 x+6 y+2 z\)
- \(3 x^{2}-5 x+7\)
- \(-a^{3}\)
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: \(25 y^{2}+15 y^{2}\).
Solution
\(25 y^{2}+15 y^{2}\) | |
Combine like terms. | \(40 y^{2}\) |
Try It 6.5
Add: \(12 q^{2}+9 q^{2}\).
Try It 6.6
Add: \(-15 c^{2}+8 c^{2}\).
Example 6.4
Subtract: \(16p−(−7p)\).
Solution
\(16p−(−7p\)) | |
Combine like terms. | \(23p\) |
Try It 6.7
Subtract: \(8 m-(-5 m)\).
Try It 6.8
Subtract: \(-15 z^{3}-\left(-5 z^{3}\right)\).
Remember that like terms must have the same variables with the same exponents.
Example 6.5
Simplify: \(c^{2}+7 d^{2}-6 c^{2}\).
Solution
\(c^{2}+7 d^{2}-6 c^{2}\) | |
Combine like terms. | \(-5 c^{2}+7 d^{2}\) |
Try It 6.9
Add: \(8 y^{2}+3 z^{2}-3 y^{2}\).
Try It 6.10
Add: \(3 m^{2}+n^{2}-7 m^{2}\).
Example 6.6
Simplify: \(u^2v+5u^2−3v^2\).
Solution
\(u^{2} v+5 u^{2}-3 v^{2}\) | |
There are no like terms to combine. | \(u^{2} v+5 u^{2}-3 v^{2}\) |
Try It 6.11
Simplify: \(m^{2} n^{2}-8 m^{2}+4 n^{2}\).
Try It 6.12
Simplify: \(p q^{2}-6 p-5 q^{2}\).
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: \(\left(5 y^{2}-3 y+15\right)+\left(3 y^{2}-4 y-11\right)\).
Solution
Identify like terms. | \(\left(\underline{\underline{5 y^{2}}}-\underline{3 y}+\underline{15}\right)+\left(\underline{\underline{3 y^{2}}}-\underline{4 y}-\underline{11}\right)\) |
Rearrange to get the like terms together. | \(\underline{5 y^{2}+3 y^{2}}-\underline{3 y-4 y}+\underline{15}-\underline{11}\) |
Combine like terms. | \(8 y^{2}-7 y+4\) |
Try It 6.13
Find the sum: \(\left(7 x^{2}-4 x+5\right)+\left(x^{2}-7 x+3\right)\).
Try It 6.14
Find the sum: \(\left(14 y^{2}+6 y-4\right)+\left(3 y^{2}+8 y+5\right)\).
Example 6.8
Find the difference: \((9w^2−7w+5)−(2w^2−4)\).
Solution
\(\left(9 w^{2}-7 w+5\right)-\left(2 w^{2}-4\right)\) | |
Distribute and identify like terms. | \(\underline{\underline{9 w^{2}}}-\underline{7 w}+\underset{\sim}{5}-\underline{\underline{2 w^{2}}}+\underset{\sim}{4}\) |
Rearrange the terms. | \(\underline{\underline{9 w^{2}-2 w^{2}}}-\underline{7 w} + \underset{\sim}{5}+\underset{\sim}{4}\) |
Combine like terms. | \(7 w^{2}-7 w+9\) |
Try It 6.15
Find the difference: \(\left(8 x^{2}+3 x-19\right)-\left(7 x^{2}-14\right)\).
Try It 6.16
Find the difference: \(\left(9 b^{2}-5 b-4\right)-\left(3 b^{2}-5 b-7\right)\).
Example 6.9
Subtract: \((c^2−4c+7) from (7c^2−5c+3)\).
Solution
Subtract \(\left(c^{2}-4 c+7\right)\) from \(\left(7 c^{2}-5 c+3\right)\). | |
\(\left(7 c^{2}-5 c+3\right)-\left(c^{2}-4 c+7\right)\) | |
Distribute and identify like terms. | \(\underline{\underline{7 c^{2}}}-\underline{5 c}+\underset{\sim}{3}-\underline{\underline{c^{2}}}+\underline{4 c}-\underset{\sim}{7}\) |
Rearrange the terms. | \(\underline{\underline{7 c^{2}-c^{2}}}-\underline{5 c+4 c}+\underset{\sim}{3}-\underset{\sim}{7}\) |
Combine like terms. | \(6 c^{2}-c-4\) |
Try It 6.17
Subtract: \(\left(5 z^{2}-6 z-2\right)\) from \(\left(7 z^{2}+6 z-4\right)\).
Try It 6.18
Subtract: \(\left(x^{2}-5 x-8\right)\) from \(\left(6 x^{2}+9 x-1\right)\).
Example 6.10
Find the sum: \((u^2−6uv+5v^2)+(3u^2+2uv)\).
Solution
\(\left(u^{2}-6 u v+5 v^{2}\right)+\left(3 u^{2}+2 u v\right)\) | |
Distribute. | \(u^{2}-6 u v+5 v^{2}+3 u^{2}+2 u v\) |
Rearrange the terms, to put like terms together. | \(u^{2}+3 u^{2}-6 u v+2 u v+5 v^{2}\) |
Combine like terms. | \(4 u^{2}-4 u v+5 v^{2}\) |
Try It 6.19
Find the sum: \(\left(3 x^{2}-4 x y+5 y^{2}\right)+\left(2 x^{2}-x y\right)\).
Try It 6.20
Find the sum: \(\left(2 x^{2}-3 x y-2 y^{2}\right)+\left(5 x^{2}-3 x y\right)\).
Example 6.11
Find the difference: \((p^2+q^2)−(p^2+10pq−2q^2)\).
Solution
\(\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)\) | |
Distribute. | \(p^{2}+q^{2}-p^{2}-10 p q+2 q^{2}\) |
Rearrange the terms, to put like terms together. | \(p^{2}-p^{2}-10 p q+q^{2}+2 q^{2}\) |
Combine like terms. | \(-10 p q^{2}+3 q^{2}\) |
Try It 6.21
Find the difference: \(\left(a^{2}+b^{2}\right)-\left(a^{2}+5 a b-6 b^{2}\right)\).
Try It 6.22
Find the difference: \(\left(m^{2}+n^{2}\right)-\left(m^{2}-7 m n-3 n^{2}\right)\).
Example 6.12
Simplify: \((a^3−a^2b)−(ab^2+b^3)+(a^2b+ab^2)\).
Solution
\(\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)\) | |
Distribute. | \(a^{3}-a^{2} b-a b^{2}-b^{3}+a^{2} b+a b^{2}\) |
Rearrange the terms, to put like terms together. | \(a^{3}-a^{2} b+a^{2} b-a b^{2}+a b^{2}-b^{3}\) |
Combine like terms. | \(a^{3}-b^{3}\) |
Try It 6.23
Simplify: \(\left(x^{3}-x^{2} y\right)-\left(x y^{2}+y^{3}\right)+\left(x^{2} y+x y^{2}\right)\).
Try It 6.24
Simplify: \(\left(p^{3}-p^{2} q\right)+\left(p q^{2}+q^{3}\right)-\left(p^{2} q+p q^{2}\right)\).
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 \(5 x^{2}-8 x+4\) when
- \(x=4\)
- \(x=−2\)
- \(x=0\)
Solution
|
|
\(5 x^{2}-8 x+4\) | |
Substitute \(4\) for \(x\). | \(5(4)^{2}-8(4)+4\) |
Simplify the exponents. | \(5 \cdot 16-8(4)+4\) |
Multiply. | \(80-32+4\) |
Simplify. | \(52\) |
|
|
\(5 x^{2}-8 x+4\) | |
Substitute \(-2\) for \(x\). | \(5(-2)^{2}-8(-2)+4\) |
Simplify the exponents. | \(5 \cdot 4-8(-2)+4\) |
Multiply. | \(20+16+4\) |
Simplify. | \(40\) |
|
|
\(5 x^{2}-8 x+4\) | |
Substitute \(0\) for \(x\). | \(5(0)^{2}-8(0)+4\) |
Simplify the exponents. | \(5 \cdot 0-8(0)+4\) |
Multiply. | \(0+0+4\) |
Simplify. | \(4\) |
Try It 6.25
Evaluate: \(3 x^{2}+2 x-15\) when
- \(x=3\)
- \(x=−5\)
- \(x=0\)
Try It 6.26
Evaluate: \(5 z^{2}-z-4\) when
- \(z=−2\)
- \(z=0\)
- \(z=2\)
Example 6.14
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 \(t=2\) seconds.
Solution
\(-16 t^{2}+250\) | |
Substitute \(t=2\). | \(-16(2)^{2}+250\) |
Simplify. | \(-16 \cdot 4+250\) |
Simplify. | \(-64+250\) |
Simplify. | \(186\) |
After 2 seconds the height of the ball is 186 feet. |
Try It 6.27
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.
Try It 6.28
The polynomial \(−16t2+250\) 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.
Example 6.15
The polynomial \(6 x^{2}+15 x y\) 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 \(x=4\) feet and \(y=6\) feet.
Solution
\(6 x^{2}+15 x y\) | |
Substitute \(x=4, y=6\) | \(6(4)^{2}+15(4)(6)\) |
Simplify. | \(6 \cdot 16+15(4)(6)\) |
Simplify. | \(96+360\) |
Simplify. | \(456\) |
The cost of producing the box is $456. |
Try It 6.29
The polynomial \(6 x^{2}+15 x y\) 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 \(x=6\) feet and \(y=4\) feet.
Try It 6.30
The polynomial \(6 x^{2}+15 x y\) 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 \(x=5\) feet and \(y=8\) feet.