(3x^3-2x^2-7x+6)/(x+1)

5 min read Jun 16, 2024

Dividing Polynomials: A Step-by-Step Guide with (3x^3 - 2x^2 - 7x + 6) / (x + 1)

In algebra, dividing polynomials is a fundamental operation that often arises in solving equations, simplifying expressions, and understanding the behavior of functions. Let's walk through the process of dividing the polynomial (3x^3 - 2x^2 - 7x + 6) by the binomial (x + 1) using polynomial long division.

1. Setting Up the Division

First, we set up the problem similar to traditional long division with the dividend (3x^3 - 2x^2 - 7x + 6) inside the division symbol and the divisor (x + 1) outside.

          _________
x + 1 | 3x^3 - 2x^2 - 7x + 6

2. Divide the Leading Terms

We begin by dividing the leading term of the dividend (3x^3) by the leading term of the divisor (x). This gives us 3x^2. Write this term above the dividend in the quotient area.

          3x^2       
x + 1 | 3x^3 - 2x^2 - 7x + 6

3. Multiply and Subtract

Next, multiply the quotient term (3x^2) by the entire divisor (x + 1) to get 3x^3 + 3x^2. Write this result below the dividend and subtract it.

          3x^2       
x + 1 | 3x^3 - 2x^2 - 7x + 6 
        -(3x^3 + 3x^2)
        ------------------
               -5x^2 - 7x

4. Bring Down the Next Term

Bring down the next term of the dividend (-7x) to form the new expression below the line.

          3x^2       
x + 1 | 3x^3 - 2x^2 - 7x + 6 
        -(3x^3 + 3x^2)
        ------------------
               -5x^2 - 7x

5. Repeat Steps 2-4

Repeat the process of dividing the leading term (-5x^2) by the divisor's leading term (x) to get -5x. Write this term in the quotient area.

          3x^2 - 5x     
x + 1 | 3x^3 - 2x^2 - 7x + 6 
        -(3x^3 + 3x^2)
        ------------------
               -5x^2 - 7x 
               -(-5x^2 - 5x)
               ------------------
                         -2x + 6

Bring down the next term (6).

          3x^2 - 5x     
x + 1 | 3x^3 - 2x^2 - 7x + 6 
        -(3x^3 + 3x^2)
        ------------------
               -5x^2 - 7x 
               -(-5x^2 - 5x)
               ------------------
                         -2x + 6

6. Final Division

Repeat the process one last time. Divide the leading term (-2x) by the divisor's leading term (x) to get -2. Write this term in the quotient area.

          3x^2 - 5x - 2
x + 1 | 3x^3 - 2x^2 - 7x + 6 
        -(3x^3 + 3x^2)
        ------------------
               -5x^2 - 7x 
               -(-5x^2 - 5x)
               ------------------
                         -2x + 6
                         -(-2x - 2)
                         --------------
                                 8

7. The Result

The process stops when the degree of the remainder (8) is less than the degree of the divisor (x + 1).

Therefore, the result of dividing (3x^3 - 2x^2 - 7x + 6) by (x + 1) is:

(3x^3 - 2x^2 - 7x + 6) / (x + 1) = 3x^2 - 5x - 2 + 8/(x + 1)

This represents the quotient (3x^2 - 5x - 2) and the remainder (8) over the divisor (x + 1).