(4x^3-3x^2-2x+1)/(x+1)

5 min read Jun 16, 2024

Polynomial Long Division: (4x^3 - 3x^2 - 2x + 1) / (x + 1)

This article will guide you through the process of dividing the polynomial 4x^3 - 3x^2 - 2x + 1 by the binomial x + 1 using polynomial long division.

Step 1: Set up the Division

Write the division problem as you would with regular long division, with the dividend (4x^3 - 3x^2 - 2x + 1) inside the division symbol and the divisor (x + 1) outside.

             ________
x + 1 | 4x^3 - 3x^2 - 2x + 1

Step 2: Divide the Leading Terms

Divide the leading term of the dividend (4x^3) by the leading term of the divisor (x). This gives us 4x^2. Write this term above the division symbol.

             4x^2 _______
x + 1 | 4x^3 - 3x^2 - 2x + 1

Step 3: Multiply the Quotient by the Divisor

Multiply the quotient (4x^2) by the divisor (x + 1). This gives us 4x^3 + 4x^2. Write this result under the dividend, aligning terms with matching powers.

             4x^2 _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2

Step 4: Subtract

Subtract the result from the previous step (4x^3 + 4x^2) from the dividend. This gives us -7x^2 - 2x.

             4x^2 _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2
        -----------
              -7x^2 - 2x

Step 5: Bring Down the Next Term

Bring down the next term from the dividend (-2x).

             4x^2 _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2
        -----------
              -7x^2 - 2x + 1

Step 6: Repeat Steps 2-5

Repeat steps 2-5 with the new polynomial (-7x^2 - 2x + 1). Divide the leading term (-7x^2) by the leading term of the divisor (x), which gives us -7x. Write this term next to 4x^2 in the quotient.

             4x^2 - 7x _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2
        -----------
              -7x^2 - 2x + 1
              -7x^2 - 7x

Subtract to get 5x + 1. Bring down the next term (1).

             4x^2 - 7x _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2
        -----------
              -7x^2 - 2x + 1
              -7x^2 - 7x
              ----------
                   5x + 1

Repeat the process. Divide 5x by x to get 5.

             4x^2 - 7x + 5 _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2
        -----------
              -7x^2 - 2x + 1
              -7x^2 - 7x
              ----------
                   5x + 1
                   5x + 5

Subtract to get -4.

             4x^2 - 7x + 5 _______
x + 1 | 4x^3 - 3x^2 - 2x + 1 
        4x^3 + 4x^2
        -----------
              -7x^2 - 2x + 1
              -7x^2 - 7x
              ----------
                   5x + 1
                   5x + 5
                   -------
                      -4

Step 7: Write the Result

The quotient is 4x^2 - 7x + 5 and the remainder is -4. We can express the result as:

(4x^3 - 3x^2 - 2x + 1) / (x + 1) = 4x^2 - 7x + 5 - 4/(x + 1)

Therefore, the division of (4x^3 - 3x^2 - 2x + 1) by (x + 1) results in a quotient of 4x^2 - 7x + 5 and a remainder of -4.