17Calculus Precalculus - Polynomial Long Division

17Calculus

Long Division of Polynomials

Although not usually covered in precalculus, long division of polynomials is an essential skill in calculus. If your instructor allows it, we recommend you use synthetic division when you can. However, long division will work in more cases than synthetic division. And you need to learn long division anyway. As usual, check with your instructor to see what they expect.

Long division is best learned first by watching someone do it and then trying it on your own own. This first video is a complete lection on the topic by one of our favorite instructors.

Prof Leonard - Long Division of Polynomials [34mins-45secs]

video by Prof Leonard

Here is a great video comparing long division to synthetic division side-by-side using an example.

Learn Math Tutorials - Synthetic Division vs. Long Division [6min-26secs]

Okay, that should be enough to refresh your memory. Try the practice problems. Remember you will get better as you go along, so practice until it becomes easy.

Practice

Unless otherwise instructed, simplify each expression using polynomial long division.

Basic

$$\displaystyle{ \frac{ x^2 + 5x + 6 }{ x + 2 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^2 + 5x + 6 }{ x + 2 } }$$ using polynomial long division.

$$x+3$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^2 + 5x + 6 }{ x + 2 } }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2678 video solution

$$x+3$$

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$$\displaystyle{ \frac{ x^2 + 3x + 5 }{ x+1 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^2 + 3x + 5 }{ x+1 } }$$ using polynomial long division.

$$\displaystyle{ x + 2 + \frac{3}{x+1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^2 + 3x + 5 }{ x+1 } }$$ using polynomial long division.

Solution

NancyPi - 2679 video solution

video by NancyPi

$$\displaystyle{ x + 2 + \frac{3}{x+1} }$$

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$$\displaystyle{ \frac{ x^3 - 1 }{ x - 1 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^3 - 1 }{ x - 1 } }$$ using polynomial long division.

$$\displaystyle{ x^2 + x + 1 }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^3 - 1 }{ x - 1 } }$$ using polynomial long division.

Solution

MySecretMathTutor - 2680 video solution

$$\displaystyle{ x^2 + x + 1 }$$

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$$\displaystyle{ \frac{2x^3 + 3x^2 - 30x + 7}{x - 3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{2x^3 + 3x^2 - 30x + 7}{x - 3} }$$ using polynomial long division.

$$\displaystyle{ 2x^2 + 9x - 3 - \frac{2}{x - 3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{2x^3 + 3x^2 - 30x + 7}{x - 3} }$$ using polynomial long division.

Solution

MrB4math - 2681 video solution

video by MrB4math

$$\displaystyle{ 2x^2 + 9x - 3 - \frac{2}{x - 3} }$$

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$$\displaystyle{ \frac{2x^3 + 8x^2 - 6x + 10}{x - 2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{2x^3 + 8x^2 - 6x + 10}{x - 2} }$$ using polynomial long division.

$$\displaystyle{ 2x^2 + 12x + 18 + \frac{46}{x - 2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{2x^3 + 8x^2 - 6x + 10}{x - 2} }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2682 video solution

$$\displaystyle{ 2x^2 + 12x + 18 + \frac{46}{x - 2} }$$

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$$\displaystyle{ \frac{x^2 + 4x - 8}{x - 2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^2 + 4x - 8}{x - 2} }$$ using polynomial long division.

$$\displaystyle{ x + 6 + \frac{4}{x-2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^2 + 4x - 8}{x - 2} }$$ using polynomial long division.

Solution

PatrickJMT - 2683 video solution

video by PatrickJMT

$$\displaystyle{ x + 6 + \frac{4}{x-2} }$$

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$$\displaystyle{ \frac{6x^4 - 9x^2 + 18}{x - 3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{6x^4 - 9x^2 + 18}{x - 3} }$$ using polynomial long division.

$$\displaystyle{ 6x^3 + 18x^2 + 45x + 145 +\frac{453}{x - 3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{6x^4 - 9x^2 + 18}{x - 3} }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2684 video solution

$$\displaystyle{ 6x^3 + 18x^2 + 45x + 145 +\frac{453}{x - 3} }$$

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$$\displaystyle{ \frac{x^2 + 2x - 7}{ x-2 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^2 + 2x - 7}{ x-2 } }$$ using polynomial long division.

$$\displaystyle{ x + 4 + \frac{1}{x-2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^2 + 2x - 7}{ x-2 } }$$ using polynomial long division.

Solution

Krista King Math - 2685 video solution

video by Krista King Math

$$\displaystyle{ x + 4 + \frac{1}{x-2} }$$

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$$\displaystyle{ \frac{x^3 + 4x + 6}{x^2 + 1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^3 + 4x + 6}{x^2 + 1} }$$ using polynomial long division.

$$\displaystyle{ x + \frac{3x+6}{x^2+1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^3 + 4x + 6}{x^2 + 1} }$$ using polynomial long division.

Solution

PatrickJMT - 2686 video solution

video by PatrickJMT

$$\displaystyle{ x + \frac{3x+6}{x^2+1} }$$

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$$\displaystyle{ \frac{6x^2 + 7x - 20}{2x + 5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{6x^2 + 7x - 20}{2x + 5} }$$ using polynomial long division.

$$3x - 4$$

Problem Statement

Simplify $$\displaystyle{ \frac{6x^2 + 7x - 20}{2x + 5} }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2687 video solution

$$3x - 4$$

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$$\displaystyle{ \frac{ 6x^4 - 30x^2 + 24 }{ 2x^2 - 8 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 6x^4 - 30x^2 + 24 }{ 2x^2 - 8 } }$$ using polynomial long division.

$$3x^2 - 3$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 6x^4 - 30x^2 + 24 }{ 2x^2 - 8 } }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2688 video solution

$$3x^2 - 3$$

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Intermediate

$$\displaystyle{ \frac{ 3x^5 + 4x^3 - 5x + 8 }{ x^2 + 3 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 3x^5 + 4x^3 - 5x + 8 }{ x^2 + 3 } }$$ using polynomial long division.

$$\displaystyle{ 3x^3 - 5x + \frac{10x+8}{x^2+3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 3x^5 + 4x^3 - 5x + 8 }{ x^2 + 3 } }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2689 video solution

$$\displaystyle{ 3x^3 - 5x + \frac{10x+8}{x^2+3} }$$

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$$\displaystyle{ \frac{x^5 + 2x^4 + x^3 - x^2 - 22x + 15}{x^2 + 2x - 3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^5 + 2x^4 + x^3 - x^2 - 22x + 15}{x^2 + 2x - 3} }$$ using polynomial long division.

$$\displaystyle{ x^3 + 4x - 9 + \frac{8x-12}{x^2+2x-3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^5 + 2x^4 + x^3 - x^2 - 22x + 15}{x^2 + 2x - 3} }$$ using polynomial long division.

Solution

The Organic Chemistry Tutor - 2690 video solution

$$\displaystyle{ x^3 + 4x - 9 + \frac{8x-12}{x^2+2x-3} }$$

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$$\displaystyle{ \frac{ 6x^3 + 10x^2 + 8 }{ 2x^2 + 1 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 6x^3 + 10x^2 + 8 }{ 2x^2 + 1 } }$$ using polynomial long division.

$$\displaystyle{ 3x + 5 + \frac{-3x+3}{2x^2+1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 6x^3 + 10x^2 + 8 }{ 2x^2 + 1 } }$$ using polynomial long division.

Solution

NancyPi - 2691 video solution

video by NancyPi

$$\displaystyle{ 3x + 5 + \frac{-3x+3}{2x^2+1} }$$

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$$\displaystyle{ \frac{ 5x^3 - 6x^2 - 28x - 2 }{ x+2 } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 5x^3 - 6x^2 - 28x - 2 }{ x+2 } }$$ using polynomial long division.

His answer is correct but it should be written in this form.
$$\displaystyle{ 5x^2 - 16x + 4 + \frac{10}{x+2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ 5x^3 - 6x^2 - 28x - 2 }{ x+2 } }$$ using polynomial long division.

Solution

MySecretMathTutor - 2692 video solution

His answer is correct but it should be written in this form.
$$\displaystyle{ 5x^2 - 16x + 4 + \frac{10}{x+2} }$$

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$$\displaystyle{ \frac{2x^3 + 9x^2 - 19x + 7}{2x - 1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{2x^3 + 9x^2 - 19x + 7}{2x - 1} }$$ using polynomial long division.

$$\displaystyle{ x^2 + 5x - 7 }$$

Problem Statement

Simplify $$\displaystyle{ \frac{2x^3 + 9x^2 - 19x + 7}{2x - 1} }$$ using polynomial long division.

Solution

ProfRobBob - 2693 video solution

video by ProfRobBob

$$\displaystyle{ x^2 + 5x - 7 }$$

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$$\displaystyle{ \frac{y^5 - 3y^2 + 20}{y-2} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{y^5 - 3y^2 + 20}{y-2} }$$ using polynomial long division.

$$y^4 + 2y^3 + 4y^2 + 5y + 10$$

Problem Statement

Simplify $$\displaystyle{ \frac{y^5 - 3y^2 + 20}{y-2} }$$ using polynomial long division.

Solution

ProfRobBob - 2694 video solution

video by ProfRobBob

$$y^4 + 2y^3 + 4y^2 + 5y + 10$$

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$$\displaystyle{ \frac{x^4 + 3x^2 - 6x - 10}{x^2 + 3x - 5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^4 + 3x^2 - 6x - 10}{x^2 + 3x - 5} }$$ using polynomial long division.

$$\displaystyle{ x^2 - 3x + 17 + \frac{-72x+75}{x^2+3x-5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^4 + 3x^2 - 6x - 10}{x^2 + 3x - 5} }$$ using polynomial long division.

Solution

Thinkwell - 2695 video solution

video by Thinkwell

$$\displaystyle{ x^2 - 3x + 17 + \frac{-72x+75}{x^2+3x-5} }$$

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$$\displaystyle{ \frac{4x^2 - 2x + 3}{x-1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{4x^2 - 2x + 3}{x-1} }$$ using polynomial long division.

$$\displaystyle{ 4x + 2 + \frac{5}{x-1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{4x^2 - 2x + 3}{x-1} }$$ using polynomial long division.

Solution

Mario's Math Tutoring - 2696 video solution

$$\displaystyle{ 4x + 2 + \frac{5}{x-1} }$$

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$$\displaystyle{ \frac{6x^2 - 3x + 1}{3x-1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{6x^2 - 3x + 1}{3x-1} }$$ using polynomial long division.

$$\displaystyle{ 2x - 1/3 + \frac{26/3}{3x-1} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{6x^2 - 3x + 1}{3x-1} }$$ using polynomial long division.

Solution

Mario's Math Tutoring - 2697 video solution

$$\displaystyle{ 2x - 1/3 + \frac{26/3}{3x-1} }$$

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$$\displaystyle{ \frac{-27x^4 + 57x^3 - 49x^2 - 37x + 17}{9x^2 + 2x - 3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{-27x^4 + 57x^3 - 49x^2 - 37x + 17}{9x^2 + 2x - 3} }$$ using polynomial long division.

$$\displaystyle{ -3x^2 + 7x - 8 - \frac{7}{9x^2+2x-3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{-27x^4 + 57x^3 - 49x^2 - 37x + 17}{9x^2 + 2x - 3} }$$ using polynomial long division.

Solution

MIP4U - 2698 video solution

video by MIP4U

$$\displaystyle{ -3x^2 + 7x - 8 - \frac{7}{9x^2+2x-3} }$$

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$$\displaystyle{ \frac{x^4 - x^2 + x - 4}{x^2 - 2x + 5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^4 - x^2 + x - 4}{x^2 - 2x + 5} }$$ using polynomial long division.

$$\displaystyle{ x^2 + 2x - 2 + \frac{-13x+6}{x^2-2x+5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^4 - x^2 + x - 4}{x^2 - 2x + 5} }$$ using polynomial long division.

Solution

PatrickJMT - 2699 video solution

video by PatrickJMT

$$\displaystyle{ x^2 + 2x - 2 + \frac{-13x+6}{x^2-2x+5} }$$

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$$\displaystyle{ \frac{ x^3 - y^3 }{ x-y } }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^3 - y^3 }{ x-y } }$$ using polynomial long division.

$$\displaystyle{ x^2 + xy + y^2 }$$

Problem Statement

Simplify $$\displaystyle{ \frac{ x^3 - y^3 }{ x-y } }$$ using polynomial long division.

Solution

Krista King Math - 2700 video solution

video by Krista King Math

$$\displaystyle{ x^2 + xy + y^2 }$$

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