Step by Step Polynomial Long Division

Step by Step Polynomial Long Division Guide

Introduction

Hi, I’m John Johnson, and today I want to show you how to do polynomial long division step by step. I’ve personally found this method to be incredibly helpful in solving polynomial equations, and I’m excited to share it with you.

What is Polynomial Long Division?

Before we dive in, let’s first define what polynomial long division is. Polynomial long division is a method used to divide two polynomials, where the degree of the dividend polynomial is greater than or equal to the degree of the divisor polynomial.

• Polynomial long division is a powerful tool in solving polynomial equations
• It is used when the degree of the dividend polynomial is greater than or equal to the degree of the divisor polynomial
• It involves dividing the terms of the dividend polynomial by the terms of the divisor polynomial

Step by Step Polynomial Long Division Guide

Now that we’ve defined what polynomial long division is, let’s dive into the step by step guide on how to do it.

Step 1: Divide the First Term of the Dividend by the First Term of the Divisor

The first step in polynomial long division is to divide the first term of the dividend by the first term of the divisor. Write the result above the line as the first term of the quotient.

Step 2: Multiply the Divisor by the First Term of the Quotient

The second step is to multiply the divisor by the first term of the quotient. Write the result under the first term of the dividend.

Step 3: Subtract the Result from Step 2 from the First Two Terms of the Dividend

The third step is to subtract the result from step 2 from the first two terms of the dividend. Write the result under the line.

Step 4: Bring Down the Next Term of the Dividend

The fourth step is to bring down the next term of the dividend. Write it next to the result from step 3.

Step 5: Repeat Steps 1-4 Until the Remainder is Zero or Has a Lower Degree than the Divisor

The fifth and final step is to repeat steps 1-4 until the remainder is zero or has a lower degree than the divisor. Write the final quotient as the result of all the quotients obtained in the previous steps.

Examples

Let’s now take a look at some examples to better understand how to do polynomial long division step by step.

Example 1

Divide x^3 + 2x^2 – 3x – 4 by x – 2

Solution:

First, we set up the problem like this:

	x - 2 | x^3 + 2x^2 - 3x - 4
	

Next, we divide the first term of the dividend by the first term of the divisor:

	x - 2 | x^3 + 2x^2 - 3x - 4
	x + 2 |
	

Then, we multiply the divisor by the first term of the quotient:

	x - 2 | x^3 + 2x^2 - 3x - 4
	x + 2 | x^2 -  x + 2
	      |------------
	      | 3x^2 - 5x
	

Next, we subtract the result from step 2 from the first two terms of the dividend:

	x - 2 | x^3 + 2x^2 - 3x - 4
	x + 2 | x^2 -  x + 2
	      |------------
	      | 3x^2 - 5x
	      | 3x^2 - 6x
	      |---------
	      |        x - 4
	

Finally, we bring down the next term of the dividend and repeat the steps until we get a remainder with a lower degree than the divisor:

	x - 2 | x^3 + 2x^2 - 3x - 4
	x + 2 | x^2 -  x + 2
	      |------------
	      | 3x^2 - 5x
	      | 3x^2 - 6x
	      |---------
	      |        x - 4
	      |        x - 2
	      |---------
	      |          -2
	

Therefore, the result of dividing x^3 + 2x^2 – 3x – 4 by x – 2 is x^2 – x + 2 with a remainder of -2.

FAQs

What is the degree of the dividend polynomial in polynomial long division?

The degree of the dividend polynomial is the highest power of the variable in the polynomial.

What is the degree of the divisor polynomial in polynomial long division?

The degree of the divisor polynomial is the highest power of the variable in the polynomial.

Why is polynomial long division useful?

Polynomial long division is useful in solving polynomial equations, especially when the degree of the dividend polynomial is greater than or equal to the degree of the divisor polynomial.

Is polynomial long division difficult to learn?

Polynomial long division may seem intimidating at first, but with practice and following the steps outlined in this guide, it can become easier to understand and apply.

What are some common mistakes to avoid when doing polynomial long division?

Some common mistakes to avoid when doing polynomial long division include forgetting to add or subtract terms, dividing the wrong terms, and not bringing down the next term of the dividend.