(x^3+5x^2+5x-3)/(x^2+3x-1)

5 min read Jun 17, 2024

Dividing Polynomials: (x^3+5x^2+5x-3) / (x^2+3x-1)

This article will guide you through the process of dividing the polynomial (x^3+5x^2+5x-3) by (x^2+3x-1). We will utilize polynomial long division, a method similar to the long division we learned in elementary school.

Step 1: Set up the division

First, write the division problem in a similar format to long division of numbers:

        ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3

Step 2: Divide the leading terms

Focus on the leading terms of both polynomials: x^3 (dividend) and x^2 (divisor).
Ask yourself: "What do I multiply x^2 by to get x^3?" The answer is x.
Write x above the line, aligning it with the x^2 term in the dividend.

        x     ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3

Step 3: Multiply and subtract

Multiply the divisor (x^2 + 3x - 1) by the term we just wrote (x):
- x * (x^2 + 3x - 1) = x^3 + 3x^2 - x
Write this result below the dividend, aligning like terms:

        x     ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3
           x^3 + 3x^2 - x 
           --------------

Subtract the two expressions. Notice that the x^3 terms cancel out:

        x     ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3
           x^3 + 3x^2 - x 
           --------------
              2x^2 + 6x - 3

Step 4: Bring down the next term

Bring down the next term (-3) from the dividend:

        x     ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3
           x^3 + 3x^2 - x 
           --------------
              2x^2 + 6x - 3

Step 5: Repeat steps 2-4

Now, focus on the new leading term of the dividend (2x^2) and the leading term of the divisor (x^2):

Ask: "What do I multiply x^2 by to get 2x^2?" The answer is 2.
Write +2 above the line:

        x + 2  ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3
           x^3 + 3x^2 - x 
           --------------
              2x^2 + 6x - 3

Multiply the divisor by 2:
- 2 * (x^2 + 3x - 1) = 2x^2 + 6x - 2
Write this below the previous result and subtract:

        x + 2  ________
x^2+3x-1 | x^3 + 5x^2 + 5x - 3
           x^3 + 3x^2 - x 
           --------------
              2x^2 + 6x - 3 
              2x^2 + 6x - 2
              --------------
                       -1

Step 6: The remainder

Since the degree of the remaining term (-1) is less than the degree of the divisor (x^2 + 3x - 1), we stop here.

The Result

The result of dividing (x^3 + 5x^2 + 5x - 3) by (x^2 + 3x - 1) is:

x + 2 - 1/(x^2 + 3x - 1)

This can also be written as:

(x^3 + 5x^2 + 5x - 3) / (x^2 + 3x - 1) = x + 2 + (-1)/(x^2 + 3x - 1)