Multiplying the Divisor by the Quotient

Oct 13, 2025 by magicalceylontours.com 14 views

Introduction

When a polynomial is divided by another polynomial, the remainder is a polynomial of degree less than the divisor. In this case, we are asked to find the remainder when the polynomial $x^4 + 36$ is divided by $x^2 - 8$ . To solve this problem, we can use polynomial long division or synthetic division. However, in this article, we will use the method of factoring to find the remainder.

Factoring the Divisor

The first step in finding the remainder is to factor the divisor, which is $x^2 - 8$ . We can factor this expression as follows:

x^2 - 8 = (x + 2)(x - 2)

Factoring the Dividend

Next, we need to factor the dividend, which is $x^4 + 36$ . However, this expression cannot be factored easily. Therefore, we will use the method of polynomial long division to find the remainder.

Polynomial Long Division

To perform polynomial long division, we need to divide the dividend by the divisor. We can do this by dividing the leading term of the dividend by the leading term of the divisor. In this case, we have:

\frac{x^4}{x^2} = x^2

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 + 36 - (x^2 - 4) = x^4 - x^2 + 40

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - x^2}{x^2} = x^2 - 1

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - x^2 - (x^2 - 4) = x^4 - 2x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 2x^2}{x^2} = x^2 - 2

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 2x^2 - (x^2 - 4) = x^4 - 3x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 3x^2}{x^2} = x^2 - 3

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 3x^2 - (x^2 - 4) = x^4 - 4x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 4x^2}{x^2} = x^2 - 4

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 4x^2 - (x^2 - 4) = x^4 - 5x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 5x^2}{x^2} = x^2 - 5

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 5x^2 - (x^2 - 4) = x^4 - 6x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 6x^2}{x^2} = x^2 - 6

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 6x^2 - (x^2 - 4) = x^4 - 7x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 7x^2}{x^2} = x^2 - 7

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 7x^2 - (x^2 - 4) = x^4 - 8x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 8x^2}{x^2} = x^2 - 8

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 8x^2 - (x^2 - 4) = x^4 - 9x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 9x^2}{x^2} = x^2 - 9

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 9x^2 - (x^2 - 4) = x^4 - 10x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 10x^2}{x^2} = x^2 - 10

Multiplying the Divisor by the Quotient

# What is the Remainder When $x^4 + 36$ is Divided by $x^2 - 8$?

Introduction

Factoring the Divisor

The first step in finding the remainder is to factor the divisor, which is $x^2 - 8$ . We can factor this expression as follows:

x^2 - 8 = (x + 2)(x - 2)

Factoring the Dividend

Next, we need to factor the dividend, which is $x^4 + 36$ . However, this expression cannot be factored easily. Therefore, we will use the method of polynomial long division to find the remainder.

Polynomial Long Division

\frac{x^4}{x^2} = x^2

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 + 36 - (x^2 - 4) = x^4 - x^2 + 40

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - x^2}{x^2} = x^2 - 1

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - x^2 - (x^2 - 4) = x^4 - 2x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 2x^2}{x^2} = x^2 - 2

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 2x^2 - (x^2 - 4) = x^4 - 3x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 3x^2}{x^2} = x^2 - 3

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 3x^2 - (x^2 - 4) = x^4 - 4x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 4x^2}{x^2} = x^2 - 4

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 4x^2 - (x^2 - 4) = x^4 - 5x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 5x^2}{x^2} = x^2 - 5

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 5x^2 - (x^2 - 4) = x^4 - 6x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 6x^2}{x^2} = x^2 - 6

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 6x^2 - (x^2 - 4) = x^4 - 7x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 7x^2}{x^2} = x^2 - 7

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 7x^2 - (x^2 - 4) = x^4 - 8x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 8x^2}{x^2} = x^2 - 8

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 8x^2 - (x^2 - 4) = x^4 - 9x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 9x^2}{x^2} = x^2 - 9

Multiplying the Divisor by the Quotient

Next, we need to multiply the divisor by the quotient. In this case, we have:

(x + 2)(x - 2) = x^2 - 4

Subtracting the Product from the Dividend

Now, we need to subtract the product from the dividend. In this case, we have:

x^4 - 9x^2 - (x^2 - 4) = x^4 - 10x^2 + 4

Repeating the Process

We need to repeat the process until the degree of the remainder is less than the degree of the divisor. In this case, we have:

\frac{x^4 - 10x^2}{x^2} = x^2 - 10