(4x^4-3x^3+5x^2-30)/(x^2+3x-4)

5 min read Jun 16, 2024

Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial 4x^4 - 3x^3 + 5x^2 - 30 by x^2 + 3x - 4.

1. Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but instead of numbers, you're working with polynomials. The goal is to find the quotient (the result of the division) and the remainder (any leftover polynomial).

2. Setting Up the Problem

First, set up the problem like a traditional long division:

             ____________
x^2 + 3x - 4 | 4x^4 - 3x^3 + 5x^2 - 30

3. Dividing the Leading Terms

Focus on the leading terms: Look at the leading term of the divisor (x^2) and the leading term of the dividend (4x^4).
Divide: What do you multiply x^2 by to get 4x^4? The answer is 4x^2. Write this above the division line.
Multiply: Multiply 4x^2 by the entire divisor (x^2 + 3x - 4): 4x^2 * (x^2 + 3x - 4) = 4x^4 + 12x^3 - 16x^2
Subtract: Subtract this result from the dividend:

             4x^2
x^2 + 3x - 4 | 4x^4 - 3x^3 + 5x^2 - 30 
             -(4x^4 + 12x^3 - 16x^2)
             ---------------------
                     -15x^3 + 21x^2

4. Bring Down the Next Term

Bring down the next term: Bring down the -30 from the dividend.
Repeat: Now focus on the new leading term (-15x^3) and the divisor's leading term (x^2).

             4x^2
x^2 + 3x - 4 | 4x^4 - 3x^3 + 5x^2 - 30 
             -(4x^4 + 12x^3 - 16x^2)
             ---------------------
                     -15x^3 + 21x^2 - 30

5. Continue the Process

Divide: What do you multiply x^2 by to get -15x^3? The answer is -15x. Write this next to the 4x^2 above the division line.
Multiply: Multiply -15x by the entire divisor: -15x * (x^2 + 3x - 4) = -15x^3 - 45x^2 + 60x
Subtract: Subtract this result from the current line:

             4x^2 - 15x
x^2 + 3x - 4 | 4x^4 - 3x^3 + 5x^2 - 30 
             -(4x^4 + 12x^3 - 16x^2)
             ---------------------
                     -15x^3 + 21x^2 - 30
                     -(-15x^3 - 45x^2 + 60x)
                     ----------------------
                              66x^2 - 60x - 30

6. Final Steps

Repeat the process: Continue bringing down terms and dividing as before. You'll get:

             4x^2 - 15x + 66
x^2 + 3x - 4 | 4x^4 - 3x^3 + 5x^2 - 30 
             -(4x^4 + 12x^3 - 16x^2)
             ---------------------
                     -15x^3 + 21x^2 - 30
                     -(-15x^3 - 45x^2 + 60x)
                     ----------------------
                              66x^2 - 60x - 30
                              -(66x^2 + 198x - 264)
                              ----------------------
                                      -258x + 234

Remainder: Since the degree of the remainder (-258x + 234) is less than the degree of the divisor (x^2 + 3x - 4), we stop here.

7. The Result

Therefore, the result of dividing (4x^4 - 3x^3 + 5x^2 - 30) by (x^2 + 3x - 4) is:

Quotient: 4x^2 - 15x + 66 Remainder: -258x + 234

We can express this as:

(4x^4 - 3x^3 + 5x^2 - 30) / (x^2 + 3x - 4) = 4x^2 - 15x + 66 + (-258x + 234) / (x^2 + 3x - 4)