4.7: Dividing Polynomials
- Page ID
- 113695
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Learning Objective
- Divide by a monomial.
Dividing by a Monomial
Recall the quotient rule for exponents: if \(x\) is nonzero and \(m\) and \(n\) are positive integers, then
\[\frac{x^{m}}{x^{n}}=x^{m-n}\]
In other words, when dividing two expressions with the same base, subtract the exponents. This rule applies when dividing a monomial by a monomial. In this section, we will assume that all variables in the denominator are nonzero.
Example \(\PageIndex{1}\)
Divide:
\(\frac{28y^{3}}{7y}\).
Solution:
Divide the coefficients and subtract the exponents of the variable \(y\).
\(\begin{aligned} \frac{28y^{3}}{7y}&=\frac{28}{7}y^{3-1} \\ &=4y^{2} \end{aligned}\)
Answer:
\(4y^{2}\)
Example \(\PageIndex{2}\)
Divide:
\(\frac{24x^{7}y^{5}}{8x^{3}y^{2}}\).
Solution:
Divide the coefficients and apply the quotient rule by subtracting the exponents of the like bases.
\(\begin{aligned} \frac{24x^{7}y^{5}}{8x^{3}y^{2}}&=\frac{24}{8}x^{7-3}y^{5-2} \\ &=3x^{4}y^{3} \end{aligned}\)
Answer:
\(3x^{4}y^{3}\)
When dividing a polynomial by a monomial, we may treat the monomial as a common denominator and break up the fraction using the following property:
\[\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\]
Applying this property results in terms that can be treated as quotients of monomials.
Example \(\PageIndex{3}\)
Divide:
\(\frac{−5x^{4}+25x^{3}−15x^{2}}{5x^{2}}\).
Solution:
Break up the fraction by dividing each term in the numerator by the monomial in the denominator and then simplify each term.
Answer:
\(-x^{2}+5x-3\cdot 1\)
Check your division by multiplying the answer, the quotient, by the monomial in the denominator, the divisor, to see if you obtain the original numerator, the dividend.
\[\color{Cerulean}{\frac{dividend}{divisor}=quotient}\]
\(or\)
\[\color{Cerulean}{dividend=divisor\cdot quotient}\]
\(\begin{aligned} 5x^{2}\cdot (-x^{2}+5x-3) &=\color{Cerulean}{5x^{2}}\color{black}{\cdot (-x^{2})+}\color{Cerulean}{5x^{2}}\color{black}{\cdot 5x-}\color{Cerulean}{5x^{2}}\color{black}{\cdot 3} \\ &=-5x^{4}+25x^{3}-15x^{2}\quad\color{Cerulean}{\checkmark} \end{aligned}\)
Example \(\PageIndex{4}\)
Divide:
\(\frac{9a^{4}b−7a^{3}b^{2}+3a^{2}b}{−3a^{2}b}\).
Solution:
Answer:
\(-3a^{2}+\frac{7}{3}ab-1\). The check is optional and is left to the reader.
Exercise \(\PageIndex{1}\)
\((16x^{5}−8x^{4}+5x^{3}+2x^{2})÷(2x^{2})\).
- Answer
-
\(8x^{3}−4x^{2}+\frac{5}{2}x+1\)
Key Takeaways
- When dividing by a monomial, divide all terms in the numerator by the monomial and then simplify each term. To simplify each term, divide the coefficients and apply the quotient rule for exponents.
Exercise \(\PageIndex{3}\) Dividing by a Monomial
Divide.
- \(\frac{81y^{5}}{9y^{2}}\)
- \(\frac{36y^{9}}{9y^{3}}\)
- \(\frac{52x^{2}y}{4xy}\)
- \(\frac{24xy^{5}}{2xy^{4}}\)
- \(\frac{25x^{2}y^{5}z^{3}}{5xyz}\)
- \(\frac{−77x^{4}y^{9}z^{2}}{2x^{3}y^{3}z}\)
- \(\frac{125a^{3}b^{2}c}{−10abc}\)
- \(\frac{36a^{2}b^{3}c^{5}}{−6a^{2}b^{2}c^{3}}\)
- \(\frac{9x^{2}+27x−3}{3}\)
- \(\frac{10x^{3}−5x^{2}+40x−15}{5}\)
- \(\frac{20x^{3}−10x^{2}+30x}{2x}\)
- \(\frac{10x^{4}+8x^{2}−6x}{24x}\)
- \(\frac{−6x^{5}−9x^{3}+3x}{−3x}\)
- \(\frac{36a^{12}−6a^{9}+12a^{5}}{−12a^{5}}\)
- \(\frac{−12x^{5}+18x^{3}−6x^{2}}{−6x^{2}}\)
- \(\frac{−49a^{8}+7a^{5}−21a^{3}}{7a^{3}}\)
- \(\frac{9x^{7}−6x^{4}+12x^{3}−x^{2}}{3x^{2}}\)
- \(\frac{8x^{9}+16x^{7}−24x^{4}+8x^{3}}{−8x^{3}}\)
- \(\frac{16a^{7}−32a^{6}+20a^{5}−a^{4}}{4a^{4}}\)
- \(\frac{5a^{6}+2a^{5}+6a^{3}−12a^{2}}{3a^{2}}\)
- \(\frac{−4x^{2}y^{3}+16x^{7}y^{8}−8x^{2}y^{5}}{−4x^{2}y^{3}}\)
- \(\frac{100a^{10}b^{30}c^{5}−50a^{20}b^{5}c^{40}+20a^{5}b^{20}c^{10}}{10a^{5}b^{5}c^{5}}\)
- Find the quotient of \(−36x^{9}y^{7}\) and \(2x^{8}y^{5}\).
- Find the quotient of \(144x^{3}y^{10}z^{2}\) and \(−12x^{3}y^{5}z\).
- Find the quotient of \(3a^{4}−18a^{3}+27a^{2}\) and \(3a^{2}\).
- Find the quotient of \(64a^{2}bc^{3}−16a^{5}bc^{7}\) and \(4a^{2}bc^{3}\).
- Answer
-
1. \(9y^{3}\)
3. \(13x\)
5. \(5xy^{4}z^{2}\)
7. \(−\frac{25}{2}a^{2}b\)
9. \(3x^{2}+9x−1\)
11. \(10x^{2}−5x+15\)
13. \(2x^{4}+3x^{2}−1\)
15. \(2x^{3}−3x+1\)
17. \(3x^{5}−2x^{2}+4x−\frac{1}{3}\)
19. \(4a^{3}−8a^{2}+5a−\frac{1}{4}\)
21. \(−4x^{5}y^{5}+2y^{2}+1\)
23. \(−18xy^{2}\)
25. \(a^{2}−6a+9\)