## 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

$$x+3$$

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

$$3x - 4$$

Log in to rate this practice problem and to see it's current rating.

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

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

$$3x^2 - 3$$

Log in to rate this practice problem and to see it's current rating.

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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.
$$\displaystyle{ 5x^2 - 16x + 4 + \frac{10}{x+2} }$$

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.