7.12: Division of a Polynomial by a Monomial
Can you divide the polynomial by the monomial? How does this relate to factoring?
\begin{align*}4e^4+6e^3-10e^2 \div 2e\end{align*}
Watch This
James Sousa: Dividing Polynomials by Monomials
Guidance
Recall that a monomial is an algebraic expression that has only one term. So, for example, \begin{align*}x\end{align*}, 8, –2, or \begin{align*}3ac\end{align*} are all monomials because they have only one term. The term can be a number, a variable, or a combination of a number and a variable. A polynomial is an algebraic expression that has more than one term.
When dividing polynomials by monomials, it is often easiest to separately divide each term in the polynomial by the monomial. When simplifying each mini-division problem, don't forget to use exponent rules for the variables. For example,
\begin{align*}\frac{8x^5}{2x^3}=4x^2\end{align*}.
Remember that a fraction is just a division problem!
Example A
What is \begin{align*}(14s^2-21s+42)\div(7)\end{align*}?
Solution: This is the same as \begin{align*}\frac{14s^2-21s+42}{7}\end{align*}. Divide each term of the polynomial numerator by the monomial denominator and simplify.
- \begin{align*}\frac{14s^4}{7}=2s^4\end{align*}
- \begin{align*}\frac{-21s}{7}=-3s\end{align*}
- \begin{align*}\frac{42}{7}=6\end{align*}
Therefore, \begin{align*}(14s^2-21s+42)\div(7)=2s^4-3s+6\end{align*}.
Example B
What is \begin{align*}\frac{3w^3-18w^2-24w}{6w}\end{align*}?
Solution: Divide each term of the polynomial numerator by the monomial denominator and simplify. Remember to use exponent rules when dividing the variables.
- \begin{align*}\frac{3w^3}{6w}=\frac{w^2}{2}\end{align*}
- \begin{align*}\frac{-18w^2}{6w}=-3w\end{align*}
- \begin{align*}\frac{-24w}{6w}=-4\end{align*}
Therefore, \begin{align*}\frac{3w^3-18w^2-24w}{6w}=\frac{w^2}{2}-3w-4\end{align*}.
Example C
What is \begin{align*}(-27a^4b^5+81a^3b^4-18a^2b^3)\div(-9a^2b)\end{align*}?
Solution: This is the same as \begin{align*}\frac{-27a^4b^5+81a^3b^4-18a^2b^3}{-9a^2b}\end{align*}. Divide each term of the polynomial numerator by the monomial denominator and simplify. Remember to use exponent rules when dividing the variables.
- \begin{align*}\frac{-27a^4b^5}{-9a^2b}=3a^2b^4\end{align*}
- \begin{align*}\frac{81 a^3b^4}{-9a^2b}=-9ab^3\end{align*}
- \begin{align*}\frac{-18a^2b^3}{-9a^2b}=2b^2\end{align*}
Therefore, \begin{align*}(-27a^4b^5+81a^3b^4-18a^2b^3) \div (-9a^2b)=3a^2b^4-9ab^3+2b^2\end{align*}.
Concept Problem Revisited
Can you divide the polynomial by the monomial? How does this relate to factoring?
\begin{align*}4e^4+6e^3-10e^2 \div 2e\end{align*}
This process is the same as factoring out a \begin{align*}2e\end{align*} from the expression \begin{align*}4e^4+6e^3-10e^2\end{align*}.
- \begin{align*}\frac{4 e^4}{2e}=2e^3\end{align*}
- \begin{align*}\frac{6e^3}{2e}=3e^2\end{align*}
- \begin{align*}\frac{-10e^2}{2e}=-5e\end{align*}
Therefore, \begin{align*}4e^4+6e^3-10e^2 \div 2e=2e^3+3e^2-5e\end{align*}.
Vocabulary
- Divisor
- A divisor is the expression in the denominator of a fraction.
- Monomial
- A monomial is an algebraic expression that has only one term. \begin{align*}x\end{align*}, 8, –2, or \begin{align*}3ac\end{align*} are all monomials because they have only one term.
- Polynomial
- A polynomial is an algebraic expression that has more than one term.
Guided Practice
Complete the following division problems.
1. \begin{align*}(3a^5-5a^4+17a^3-9a^2)\div(a)\end{align*}
2. \begin{align*}(-40n^3-32n^7+88n^{11}+8n^2)\div(8n^2)\end{align*}
3. \begin{align*}\frac{16m^6-12m^4+4m^2}{4m^2}\end{align*}
Answers:
1. \begin{align*}(3a^5-5a^4+17a^3-9a^2) \div (a)=3a^4-5a^3+17a^2-9a\end{align*}
2. \begin{align*}(-40n^3-32n^7+88n^{11}+8n^2)\div(8n^2)=-5n-4n^5+11n^9+1\end{align*}
3. \begin{align*}\frac{(16m^6-12m^4+4m^2)}{(4m^2)}=4m^4-3m^2+1\end{align*}
Practice
Complete the following division problems.
- \begin{align*}(6a^3+30a^2+24a) \div 6\end{align*}
- \begin{align*}(15b^3+20b^2+5b) \div 5\end{align*}
- \begin{align*}(12c^4+18c^2+6c) \div 6c\end{align*}
- \begin{align*}(60d^{12}+90d^{11}+30d^8) \div 30d\end{align*}
- \begin{align*}(33e^7+99e^3+22e^2) \div 11e\end{align*}
- \begin{align*}(-8a^4+8a^2) \div (-4a)\end{align*}
- \begin{align*}(-3b^4+6b^3-30b^2+15b) \div (-3b)\end{align*}
- \begin{align*}(-40c^{12}-20c^{11}-25c^9-30c^3) \div 5c^2\end{align*}
- \begin{align*}(32d^{11}+16d^7+24d^4-64d^2) \div 8d^2\end{align*}
- \begin{align*}(14e^{12}-18e^{11}-12e^{10}-18e^7) \div -2e^5\end{align*}
- \begin{align*}(18a^{10}-9a^8+72a^7+9a^5+3a^2) \div 3a^2\end{align*}
- \begin{align*}(-24b^9+42b^7+42b^6) \div -6b^3\end{align*}
- \begin{align*}(24c^{12}-42c^7-18c^6) \div -2c^5\end{align*}
- \begin{align*}(14d^{12}+21d^9+42d^7) \div -7d^4\end{align*}
- \begin{align*}(-40e^{12}+30e^{10}-10e^4+30e^3+80e) \div -10e^2\end{align*}
Denominator
The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. has denominator .Dividend
In a division problem, the dividend is the number or expression that is being divided.divisor
In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression , 6 is the divisor and 152 is the dividend.Polynomial long division
Polynomial long division is the standard method of long division, applied to the division of polynomials.Rational Expression
A rational expression is a fraction with polynomials in the numerator and the denominator.Rational Root Theorem
The rational root theorem states that for a polynomial, , where are integers, the rational roots can be determined from the factors of and . More specifically, if is a factor of and is a factor of , then all the rational factors will have the form .Remainder Theorem
The remainder theorem states that if , then is the remainder when dividing by .Synthetic Division
Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.Image Attributions
Here you'll learn how to divide a polynomial by a monomial.
Concept Nodes:
Denominator
The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. has denominator .Dividend
In a division problem, the dividend is the number or expression that is being divided.divisor
In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression , 6 is the divisor and 152 is the dividend.Polynomial long division
Polynomial long division is the standard method of long division, applied to the division of polynomials.Rational Expression
A rational expression is a fraction with polynomials in the numerator and the denominator.Rational Root Theorem
The rational root theorem states that for a polynomial, , where are integers, the rational roots can be determined from the factors of and . More specifically, if is a factor of and is a factor of , then all the rational factors will have the form .Remainder Theorem
The remainder theorem states that if , then is the remainder when dividing by .Synthetic Division
Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.