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6. Polynomials & Factoring
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United States of AmericaTennessee
Algebra 1

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

A1.A.APR.A.1

Add, subtract, and multiply polynomials. Use these operations to demonstrate that polynomials form a closed system that adhere to the same properties of operations as the integers.

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP4

Model with mathematics.

A1.MP5

Use appropriate tools strategically.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

A1.MP8

Look for and express regularity in repeated reasoning.

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