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6. Polynomials & Factoring
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United States of AmericaFL
Grade 912

6.04 Dividing polynomials by a monomial

Lesson
Practice
Lesson

Concept summary

When dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial then simplify each individual fraction using the rules of exponents.

Dividing a polynomial by a monomial

\dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}

Note: Final answers are usually written without any negative exponents.

Worked examples

Example 1

Simplify the following: \dfrac{3 x^{5} + 4 x^{2}}{x}

Solution

\displaystyle \dfrac{3 x^{5} + 4 x^{2}}{x}\displaystyle =\displaystyle \dfrac{3 x^{5}}{x} + \dfrac{4 x^{2}}{x}Divide each term by x
\displaystyle =\displaystyle 3x^{4} + 4xSimplify

Since there are no negative exponents and the expression is already in standard form, the final answer is 3x^{4} + 4x.

Reflection

We can check the answer by multiplying it with the monomial in the denominator. The product should be the numerator in the original expression.

\displaystyle x \left(3x^{4} + 4x\right)\displaystyle =\displaystyle 3 x^{5} + 4 x^{2}Check

Example 2

Simplify the following: \dfrac{6 y^{3} - 15 y^{2} + 24y}{3y}

Solution

\displaystyle \dfrac{6 y^{3} - 15 y^{2} + 24y}{3y}\displaystyle =\displaystyle \dfrac{6 y^{3}}{3y} - \dfrac{15 y^{2}}{3y} + \dfrac{24 y}{3y}Divide each term by 3y
\displaystyle =\displaystyle 2y^{2} - 5y + 8Simplify

Since there are no negative exponents and the expression is already in standard form, the final answer is 2y^{2} - 5y + 8.

Reflection

We can check the answer by multiplying it with the monomial in the denominator. The product should be the numerator in the original expression.

\displaystyle 3y \left(2y^{2} - 5y + 8\right)\displaystyle =\displaystyle 6 y^{3} - 15 y^{2} + 24yCheck

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.1.4

Divide a polynomial expression by a monomial expression with rational number coefficients.

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