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6. Polynomials and Factoring
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Algebra I

6.03 Dividing polynomials

Lesson
Practice

Divide by a monomial

To divide a polynomial by a monomial, we must break the polynomial (dividend or numerator) into the product of its factors. Then, simplify the common factors from the dividend or numerator and divisor or denominator.

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.

Examples

Example 1

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

Worked Solution
Create a strategy

Apply the rule \dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}.

Apply the idea
\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.

Reflect and check

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}

Worked Solution
Create a strategy

Apply the rule \dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}.

Apply the idea
\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.

Reflect and check

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

Example 3

The triangle shown below has an area of 13n^3+11n^2+29n.

Find a simplified polynomial expression for its height.

Triangle showing its perpendicular height and base of n units.
Worked Solution
Create a strategy

Substitute the expressions into the area of triangle formula A=\dfrac 12 bh.

Apply the idea
\displaystyle A\displaystyle =\displaystyle \dfrac 12 bhWrite the area of triangle formula
\displaystyle 13n^3+11n^2+29n\displaystyle =\displaystyle \dfrac 12 \times n \times hSubstitute A=13n^3+11n^2+29n and b=n
\displaystyle \left(13n^3+11n^2+29n\right) \times 2\displaystyle =\displaystyle \left(\dfrac 12 \times n \times h\right) \times 2Multiply both sides by 2
\displaystyle 26n^3+22n^2+58n\displaystyle =\displaystyle n hEvaluate the multiplication
\displaystyle \dfrac{26n^3+22n^2+58n}{n}\displaystyle =\displaystyle \dfrac{n h}{n}Divide both sides by n
\displaystyle 26n^2+22n+58\displaystyle =\displaystyle hEvaluate the division
\displaystyle h \displaystyle =\displaystyle 26n^2+22n+58 \text{ units}Make h the subject
Idea 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.

Outcomes

A.2b

Perform operations on polynomials, including adding, subtracting, multiplying, and dividing polynomials

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