Polynomial Operations and Theorems

Polynomial Operations

Polynomials can be added, subtracted, multiplied, factored, and divided.

A polynomial in one variable is the sum or difference of terms of the form axnax^n, where aa is a real number and nn is a nonnegative integer. Polynomials with one, two, or three terms have their own names.

  • A monomial is a polynomial expression consisting of one term. An example of a monomial is 3x53x^5.
  • A binomial is a polynomial expression consisting of two terms. An example of a monomial is 4x314x^3-1.
  • A trinomial is a polynomial expression consisting of three terms. An example of a monomial is x22x8x^2-2x-8.
When written in standard form, an0a_n\neq 0, the terms of a polynomial decrease in degree as they are written from left to right:
anxn+an1xn1+an2xn2+...+a1x+a0a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} +...+a_{1}x + a_{0}
The degree of a term, or degree of a monomial, is the exponent of the variable. A constant term has degree zero. A variable with no exponent has degree one. The degree of a polynomial is the greatest degree of a term in the polynomial. In standard form, the degree of the polynomial is the degree of the first term. The leading coefficient of a polynomial is the coefficient of the first term of a polynomial written in standard form, which places the term with the highest degree first.

Adding and Subtracting Polynomials

Polynomials are added or subtracted by combining like terms.
Polynomials can be simplified in a variety of ways, including addition or subtraction. To add or subtract polynomials, combine like terms. Like terms are terms that have the same variable (or variables) with the same exponents.
Step-By-Step Example
Add Polynomials
Simplify the polynomial:
5x2+2xy+1+3x24-5x^2 + 2xy + 1 + 3x^2 - 4
Step 1
Identify like terms.
5x2+2xy+1+3x24-{\color{#c42126}{5x^2}} + 2xy + {\color{#0047af}{1}} + {\color{#c42126}{3x^2}} - {\color{#0047af}{4}}
Step 2
Write and group like terms together.
(5x2+3x2)+2xy+(14)(-{\color{#c42126}{5x^2}} + {\color{#c42126}{3x^2}}) + 2xy + ({\color{#0047af}{1}} - {\color{#0047af}{4}})
Solution
Add the like terms.
(5x2+3x2)+2xy+(14)(2x2)+2xy+(3)2x2+2xy3\begin{gathered}(-{\color{#c42126}{5x^2}} + {\color{#c42126}{3x^2}}) + 2xy + ({\color{#0047af}{1}} - {\color{#0047af}{4}})\\(-{\color{#c42126}{2x^2}})+2xy+ (-{\color{#0047af}{3}})\\ -2x^2+2xy-3\end{gathered}
Step-By-Step Example
Subtract Polynomials
Simplify the polynomial:
4x2(x2+3x5)4x^2 - (x^2+3x-5)
Step 1
Distribute the negative sign to each term in the parentheses.
4x2(x2+3x5)4x2x23x+5\begin{gathered}4x^2 - (x^2+3x-5)\\4x^2-x^2-3x+5\end{gathered}
Step 2
Identify like terms.
4x2x23x+5{\color{#c42126}{4x^2}}-{\color{#c42126}{x^2}}-3x+5
Solution
Add the coefficients of the like terms.
4x2x23x+53x23x+5\begin{gathered}{\color{#c42126}{4x^2}}-{\color{#c42126}{x^2}}-3x+5\\{\color{#c42126}{3x^2}}-3x+5\end{gathered}

Multiplying Polynomials

Polynomials are multiplied by using the distributive property and properties of exponents.

Two common methods for multiplying polynomials include using different types of properties, including the distributive property and properties of exponents.

To use the distributive property for any two polynomials, distribute the first polynomial to each term in the second polynomial. Repeat this process of distribution as needed, depending on the number of terms in the polynomials. The goal is to multiply each term in the first polynomial by each term in the second polynomial.

To use the product of powers property, multiply two powers with the same base by adding the exponents.
xmxn=xm+nx^{m} \cdot x^{n} = x^{m+n}
To multiply a monomial by a monomial, multiply the coefficients and add the exponents in the variable factors. For example:
5x2x2(52)x1+210x3\begin{gathered}5x \cdot 2x^2\\ (5\cdot 2)x^{1+2} \\ 10x^3\end{gathered}
When multiplying two binomials, the mnemonic FOIL (First, Outside, Inside, Last) provides a methodical way of finding each term in the product. This method only works for binomials, and not any other polynomials.
Step-By-Step Example
Multiply a Monomial by a Polynomial
Multiply the expression:
3x(x22x+3)3x(x^2-2x+3)
Step 1
Distribute the monomial to each term in the parentheses.
3x(x22x+3)3x(x2)3x(2x)+3x(3)\begin{gathered}3x(x^2-2x+3)\\3x(x^2)-3x(2x)+3x(3)\end{gathered}
Step 2
In each term, multiply the coefficients and add the exponents in the variable factors.
3x(x2)3x(2x)+3x(3)(31)x1+2(32)x1+1+(33)x\begin{gathered}3x(x^2)-3x(2x)+3x(3)\\(3\cdot 1)x^{1+2}-(3\cdot 2)x^{1+1}+(3\cdot 3)x\end{gathered}
Solution
Simplify the coefficients and exponents.
(31)x1+2(32)x1+1+(33)x3x36x2+9x\begin{gathered}(3\cdot 1)x^{1+2}-(3\cdot 2)x^{1+1}+(3\cdot 3)x\\3x^3-6x^2+9x\end{gathered}
Step-By-Step Example
Multiply a Binomial by a Binomial
Multiply the expression:
(2x1)(x+3)(2x-1)(x+3)
Step 1
Use the mnemonic FOIL (First, Outside, Inside, and Last terms) to distribute each term of the first factor to each term in the second factor.
(2x1)(x+3)(2x)(x)First+(2x)(3)Outside+(1)(x)Inside+(1)(3)Last\begin{gathered}(2x-1)(x+3)\\\overbrace{(2x)(x)}^\text{First}+\overbrace{(2x)(3)}^\text{Outside}+\overbrace{(-1)(x)}^\text{Inside}+\overbrace{(-1)(3)}^\text{Last}\end{gathered}
Step 2
In each term, multiply the coefficients and add the exponents in the variable factors.
(2x)(x)+(2x)(3)+(1)(x)+(1)(3)(21)x1+1+(23)x+(11)x+(3)2x2+6xx3\begin{gathered}(2x)(x)+(2x)(3)+(-1)(x)+(-1)(3)\\(2\cdot1)x^{1+1}+(2\cdot3)x+(-1\cdot1)x+(-3)\\2x^2+6x-x-3\end{gathered}
Solution
Combine like terms.
2x2+6xx32x2+5x3\begin{gathered}2x^2+{\color{#c42126}{6x-x}}-3\\2x^2+{\color{#c42126}{5x}}-3\end{gathered}
Step-By-Step Example
Multiply a Polynomial by a Polynomial
Multiply the expression:
(x32x2+1)(3x24x+7)(x^3-2x^2+1)(3x^2-4x+7)
Step 1
Distribute the first polynomial to each term in the second polynomial.
(x32x2+1)(3x24x+7)(3x2)(x32x2+1)First(4x)(x32x2+1)Second+(7)(x32x2+1)Third\begin{gathered}{\color{#c42126}{(x^3-2x^2+1)}}(3x^2-4x+7)\\\overbrace{(3x^2){\color{#c42126}{(x^3-2x^2+1)}}}^{\text{First}}-\overbrace{(4x){\color{#c42126}{(x^3-2x^2+1)}}}^{\text{Second}}+\overbrace{(7){\color{#c42126}{(x^3-2x^2+1)}}}^{\text{Third}}\end{gathered}
Step 2
Write subtraction as addition of the opposite.
(3x2)(x32x2+1)First(4x)(x32x2+1)Second+(7)(x32x2+1)Third(3x2)(x32x2+1)First+(4x)(x32x2+1)Second+(7)(x32x2+1)Third\begin{gathered}\overbrace{(3x^2)(x^3-2x^2+1)}^{\text{First}}{\color{#c42126}{-}}\overbrace{{\color{#c42126}{(4x)}}(x^3-2x^2+1)}^{\text{Second}}+\overbrace{(7)(x^3-2x^2+1)}^{\text{Third}}\\\overbrace{(3x^2)(x^3-2x^2+1)}^{\text{First}}{\color{#c42126}{+}}\overbrace{{\color{#c42126}{(-4x)}}(x^3-2x^2+1)}^{\text{Second}}+\overbrace{(7)(x^3-2x^2+1)}^{\text{Third}}\end{gathered}
Step 3
Distribute each monomial to the terms in the parentheses.
(3x2)(x32x2+1)First+(4x)(x32x2+1)Second+(7)(x32x2+1)Third\overbrace{{\color{#c42126}{(3x^2)}}(x^3-2x^2+1)}^{\text{First}}+\overbrace{{\color{#c42126}{(-4x)}}(x^3-2x^2+1)}^{\text{Second}}+\overbrace{{\color{#c42126}{(7)}}(x^3-2x^2+1)}^{\text{Third}}
First: (3x2)(x3)(3x2)(2x2)+(3x2)(1){\color{#c42126}{(3x^2)}}(x^3)-{\color{#c42126}{(3x^2)}}(2x^2)+{\color{#c42126}{(3x^2)}}(1)

Second: (4x)(x3)(4x)(2x2)+(4x)(1){\color{#c42126}{(-4x)}}(x^3)-{\color{#c42126}{(-4x)}}(2x^2)+{\color{#c42126}{(-4x)}}(1)

Third: (7)(x3)(7)(2x2)+(7)(1){\color{#c42126}{(7)}}(x^3)-{\color{#c42126}{(7)}}(2x^2)+{\color{#c42126}{(7)}}(1)

Step 4
In each term, multiply the coefficients and add the exponents in the variable factors.
3x2+36x2+2+3x2First+(4x1+3)+(8x1+2)+(4x)Second+7x314x2+7Third3x56x4+3x2First+(4x4)+8x3+(4x)Second+7x314x2+7Third\begin{gathered}\overbrace{3x^{2+3}-6x^{2+2}+3x^{2}}^{\text{First}}+\overbrace{(-4x^{1+3})+(8x^{1+2})+(-4x)}^{\text{Second}}+\overbrace{7x^{3}-14x^{2}+7}^{\text{Third}}\\\overbrace{3x^5-6x^4+3x^2}^{\text{First}}+\overbrace{(-4x^4)+8x^3+(-4x)}^{\text{Second}}+\overbrace{7x^3-14x^2+7}^{\text{Third}} \end{gathered}
Solution
Combine like terms.
3x56x4+3x2+(4x4)+8x3+(4x)+7x314x2+73x510x4+15x311x24x+7\begin{gathered}3x^5-{\color{#c42126}{6x^4}}+3x^2+(-{\color{#c42126}{4x^4}})+{\color{#0047af}{8x^3}}+(-4x)+{\color{#0047af}{7x^3}}-14x^2+7\\3x^5-{\color{#c42126}{10x^4}}+{\color{#0047af}{15x^3}}-11x^2-4x+7\end{gathered}

Factoring Polynomials

Polynomials are factored by using a variety of techniques, including grouping and formulas.
Factoring is the process of writing a number or algebraic expression as a product. The factored form of a polynomial may help when graphing a function that has the polynomial as its algebraic rule. There are different techniques for factoring polynomials.

Polynomial Factoring Methods

Method Description
Greatest common factor (GCF) When each term has a common factor, divide the greatest common factor from each term.
Factor by grouping This method is most commonly used when there are four terms. Group the terms into two pairs. Then factor out the GCF of each pair of terms. The resulting terms have a common factor that can be factored out.
Factor a trinomial Many trinomials of the form ax2+bx+cax^2+bx+c can be written as the product of two binomials. Find two numbers that are factors of acac and have a sum of bb. Use these numbers to write the binomial factors or to write a four-term polynomial and continue with factoring by grouping.
Perfect square trinomial Some trinomials fit a pattern:
a2+2ab+b2=(a+b)2a22ab+b2=(ab)2\begin{gathered}a^2+2ab+b^2=(a+b)^2\\a^2-2ab+b^2=(a-b)^2\end{gathered}
Difference of squares Some binomials fit a pattern with two squared terms:
a2b2=(a+b)(ab)a^2-b^2=(a+b)(a-b)
Sum or difference of cubes Some binomials fit a pattern with two cubed terms:
a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)\begin{gathered}a^3+b^3=(a+b)(a^2-ab+b^2)\\a^3-b^3=(a-b)(a^2+ab+b^2)\end{gathered}

Step-By-Step Example
Factor by Grouping
Factor the expression:
2x210x+3x152x^2-10x+3x-15
Step 1
Group the terms into two pairs.
2x210x+3x15(2x210x)+(3x15)\begin{gathered}2x^2-10x+3x-15\\(2x^2-10x)+(3x-15)\end{gathered}
Step 2
Factor out the GCF (greatest common factor) of each pair of terms.
2x(x5)+3(x5)2x(x-5)+3(x-5)
Solution
Factor out the GCF from the resulting terms.
(x5)(2x+3)(x-5)(2x+3)
Step-By-Step Example
Factor by Recognizing a Pattern
Factor the binomial:
8x3278x^3-27
Step 1

Identify whether the binomial fits a pattern.

The binomial is a difference of cubes:
a3b3a^3-b^3
Determine aa and bb in the binomial by identifying the cube root of each term.
8x3278x332732x3\begin{gathered}8x^3-27\\\sqrt[\scriptsize{3}]{8x^3}-\sqrt[\scriptsize{3}]{27}\\2x-3\end{gathered}
So, the values for aa and bb are a=2xa=2x and b=3b=3.
Step 2
Use the pattern for the difference of cubes to rewrite the binomial:
a3b3=(ab)(a2+ab+b2)8x327=(2x3)[(2x)2+(2x)(3)+32]\begin{aligned}a^3-b^3&=(a-b)(a^2+ab+b^2)\\8x^3-27&=(2x-3)[(2x)^2+(2x)(3)+3^2]\end{aligned}
Solution
Simplify the second factor.
8x327=(2x3)[(2x)2+(2x)(3)+32]8x327=(2x3)(4x2+6x+9)\begin{aligned}8x^3-27&=(2x-3)[(2x)^2+(2x)(3)+3^2]\\8x^3-27&=(2x-3)(4x^2+6x+9)\end{aligned}

Dividing Polynomials

Polynomials are divided by factoring or using algorithms for long division or synthetic division.

There are different techniques for dividing polynomials, depending on the polynomials that are given. The combination of monomials and polynomials will determine which technique to use. To divide by a monomial, divide each term of the dividend by the monomial. To divide by a polynomial, start by trying to factor the dividend and divisor.

The quotient of powers property states: To divide two powers with the same base, subtract the exponents.
xmxn=xmn\frac {x^{m}}{x^{n} } = x^{m-n}
Step-By-Step Example
Divide a Polynomial by a Monomial
Divide the expression:
8x32x2+4x2x\frac{8x^3-2x^2+4x}{2x}
Step 1
Divide each term in the numerator by the denominator.
8x32x2+4x2x8x32x2x22x+4x2x\begin{gathered}\frac{8x^3-2x^2+4x}{2x}\\\frac{8x^3}{2x}-\frac{2x^2}{2x}+\frac{4x}{2x}\end{gathered}
Step 2
Divide the coefficients. In the variable factors, subtract the exponent in the denominator from the exponent in the numerator.
8x32x2x22x+4x2x4x311x21+2x11\begin{gathered}\frac{8x^3}{2x}-\frac{2x^2}{2x}+\frac{4x}{2x}\\4x^{3-1}-1x^{2-1}+2x^{1-1}\end{gathered}
Solution
Simplify the exponents. Let x0=1x^0=1.
4x311x21+2x114x2x1+2x04x2x+(2)(1)4x2x+2\begin{gathered}4x^{3-1}-1x^{2-1}+2x^{1-1}\\4x^{2}-x^{1}+2x^{0}\\4x^{2}-x+(2)(1)\\4x^2-x+2\end{gathered}
Step-By-Step Example
Divide a Polynomial by a Polynomial
Divide the polynomial:
3x214x5x24x5\frac{3x^2-14x-5}{x^2-4x-5}
Step 1
Factor the numerator and denominator.
3x214x5x24x5(3x+1)(x5)(x5)(x+1)\begin{gathered}\frac{3x^2-14x-5}{x^2-4x-5}\\\frac{(3x+1)(x-5)}{(x-5)(x+1)}\end{gathered}
Step 2
Simplify.
(3x+1)(x5)(x5)(x+1)\frac{(3x+1)\cancel{(x-5)}}{\cancel{(x-5)}(x+1)}
Solution
3x214x5x24x5=3x+1x+1\frac{3x^2-14x-5}{x^2-4x-5}=\frac{3x+1}{x+1}

When factoring polynomials to divide does not work, one of the strategies that can be used is long division. The steps of long division with polynomials are similar to the steps of long division with whole numbers. This method of polynomial division can be used to divide a polynomial with a higher degree by any polynomial with a lower degree, even if the divisor is not a factor of the dividend. Before beginning, check that both the dividend and the divisor are written in standard form.
Step-By-Step Example
Divide Polynomials by Using Long Division
Divide the expression:
(2x316x+7)÷(x+3)(2x^3-16x+7) \div (x+3)
Step 1
Set up the problem for long division. Include a term for each degree of the variable, even if the coefficient of the term is zero.
x+3)2x3+0x216x+7x+3\;\overline{)\;2x^3+0x^2-16x+7}
Step 2
Divide the first term in the dividend by the first term in the divisor. Write the result above the first term in the dividend.
x+3)2x2x+3)2x3+0x216x+7\begin{aligned}&\phantom{x+3)}\;\;{\color{#c42126}{2x^{2}}}\\&{\color{#c42126}{x}}+3\;\overline{)\;{\color{#c42126}{2x^{3}}}+0x^2-16x+7}\end{aligned}
Step 3
Multiply the result by the divisor. Write the product under the dividend, and subtract. Then bring down the next term from the dividend.
x+3)2x2x+3)2x3+0x216x+7x+(2x3+6x2)6xx+3)2x3)6x216x\begin{aligned}&\phantom{x+3)}\;\;{\color{#c42126}{2x^{2}}}\\&{\color{#c42126}{x+3}}\;\overline{)\;2x^3+0x^2-16x+7}\\&\phantom{x+}\;-\underline{({\color{#c42126}{2x^3+6x^{2}}})\phantom{-}\downarrow\phantom{6x}}\\&\phantom{x+3\;)2x^3)}-6x^2-16x\end{aligned}
Step 4
Repeat the process by dividing the first term in the expression obtained after subtracting by the first term in the divisor. Write the result above the second term in the dividend.
x+3)2x26xx+3)2x3+0x216x+7x+(2x3+6x2)6xx+3)2x3)6x216x\begin{aligned}&\phantom{x+3)}\;\;2x^2{\color{#c42126}{\;-\;\;\;6x}}\\&{\color{#c42126}{x}}+3\;\overline{)\;2x^3+0x^2-16x+7}\\&\phantom{x+}\;-\underline{(2x^3+6x^2)\phantom{-}\downarrow\phantom{6x}}\;\;\;\\&\phantom{x+3\;)2x^3)}{\color{#c42126}{\;-\;6x^{2}}}-16x\end{aligned}
Step 5
Multiply the second result by the divisor. Write the product under the last expression obtained from Step 4. Then, subtract and bring down the next term from the dividend.
x+3)2x26xx+3)2x3+0x216x+7x+(2x3+6x2)6xx+3)2x3)6x216xx+3)2(6x218x)x+3)2x3+0x212x+7\begin{aligned}&\phantom{x+3)}\;\;2x^2{\color{#c42126}{\;\;-\;6x}}\\&{\color{#c42126}{x+3}}\;\overline{)\;2x^3+0x^2-16x+7}\\&\phantom{x+}\;-\underline{(2x^3+6x^2)\phantom{-}\downarrow\phantom{6x}}\;\;\;\downarrow\\&\phantom{x+3\;)2x^3)}-6x^2-16x\;\;\;\;\;\downarrow\\&\phantom{x+3)\;2}-\underline{({\color{#c42126}{-\;6x^2-18x}})\;\;\;\downarrow}\\&\phantom{x+3\;\overline{)\;2x^3+0x^2-1}}2x+7\end{aligned}
Step 6
Continue the process for the last term in the dividend.
x+3)2x26x2+2x+3)2x3+0x216x+7x+(2x3+6x2)6xx+3)2x3)6x216xx+3)2(6x218x)x+3)2x3+0x212x+7x+3)2x3+0x2(2x+6)x+3)2x3+0x2(2x+1\begin{aligned}&\phantom{x+3)}\;\;2x^2-6x\phantom{^2}+{\color{#c42126}{2}}\\&{\color{#c42126}{x+3}}\;\overline{)\;2x^3+0x^2-16x+7}\\&\phantom{x+}\;-\underline{(2x^3+6x^2)\phantom{-}\downarrow\phantom{6x}}\;\;\;\downarrow\\&\phantom{x+3\;)2x^3)}-6x^2-16x\;\;\;\;\;\downarrow\\&\phantom{x+3)\;2}-\underline{(-6x^2-18x)\;\;\;\;\downarrow}\\&\phantom{x+3\;\overline{)\;2x^3+0x^2-1}}2x+7\\&\phantom{x+3\;\overline{)\;2x^3+0x^2}}-\underline{({\color{#c42126}{2x+6}})}\\&\phantom{x+3\;\overline{)\;2x^3+0x^2}-\underline{(2x+\;\;\;\;}}1\end{aligned}
Solution
When the degree of the result of subtraction is smaller than the degree of the divisor, stop dividing. Write the remainder as a fraction with the divisor as the denominator.
2x26x+2+1x+32x^2-6x+2+\frac{1}{x+3}

Synthetic division is another strategy to use when factoring polynomials to divide does not work. Synthetic division is a process of long division of polynomials where only the coefficients and constants are recorded. The divisor must be a linear factor with a coefficient of 1. Before beginning, check that the dividend is written in standard form.
Step-By-Step Example
Divide Polynomials by Using Synthetic Division
Divide the expression:
(2x25x+7)÷(x+3)(2x^2-5x+7) \div (x+3)
Step 1
Set up the problem for synthetic division. For the dividend, include the coefficient for each degree of the variable, even if the coefficient is zero. For the divisor, use the opposite of the constant term.
Step 2
Bring down the leading coefficient.
Step 3
Multiply –3 by the 2 at the bottom, and place the product under –5. Add the values in that column, and write the sum below it.
Step 4
Repeat. Multiply –3 by –11, and place the product under 7. Add the values in that column, and write the sum below it.
Solution

Use the values in the bottom row as coefficients for each term in the quotient.

The degree of the quotient will be one less than the degree of the original dividend. After the constant term, write the remainder as a fraction with the divisor as the denominator.
2x11+40x+32x-11+\frac{40}{x+3}