To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. To do this, use the quotient rule and divide coefficients and subtract exponents with the same base.

Dividing a polynomial by a monomial

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

Polynomial division can be modeled with algebra tiles.

The image show an area model with algebra tiles. Ask your teacher for more information.

Create an area model where:

The tiles on the inside add up to the dividend (numerator).
The tiles on one side add up to the divisor (denominator).
The sum of the tiles along the other side must be the quotient (the result of the division).

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} + 4x	Simplify

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

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 + 8	Simplify

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} + 24y

Check

Example 3

The triangle shown 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 bh	Write the area of triangle formula
\displaystyle 13n^3+11n^2+29n	\displaystyle =	\displaystyle \dfrac 12 \cdot n \cdot h	Substitute A=13n^3+11n^2+29n and b=n
\displaystyle \left(13n^3+11n^2+29n\right) \cdot 2	\displaystyle =	\displaystyle \left(\dfrac 12 \cdot n \cdot h\right) \cdot 2	Multiply both sides by 2
\displaystyle 26n^3+22n^2+58n	\displaystyle =	\displaystyle n h	Evaluate 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 h	Evaluate the division
\displaystyle h	\displaystyle =	\displaystyle 26n^2+22n+58 \text{ units}	Symmetric property of equality

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.EO.2

The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2d

Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.

6.03 Divide polynomials by a monomial

Ideas

Divide by a monomial

Examples

Example 1

Example 2

Example 3

Outcomes

A.EO.2

A.EO.2d

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