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

5 min read Jun 16, 2024

Performing Polynomial Long Division: (2x^3 + 4x^2 - 5) / (x + 3)

Polynomial long division is a method used to divide polynomials, much like long division is used for integers. Let's explore how to divide (2x^3 + 4x^2 - 5) by (x + 3).

Step 1: Set up the Division

        _________
x + 3 | 2x^3 + 4x^2 - 5

Step 2: Divide the Leading Terms

The leading term of the divisor (x + 3) is x.
The leading term of the dividend (2x^3 + 4x^2 - 5) is 2x^3.
2x^3 / x = 2x^2. Write this above the dividend.

        2x^2 ______
x + 3 | 2x^3 + 4x^2 - 5

Step 3: Multiply the Quotient by the Divisor

Multiply the quotient (2x^2) by the divisor (x + 3): 2x^2 (x + 3) = 2x^3 + 6x^2.

Step 4: Subtract

Subtract the result (2x^3 + 6x^2) from the dividend (2x^3 + 4x^2 - 5):

        2x^2 ______
x + 3 | 2x^3 + 4x^2 - 5 
        -(2x^3 + 6x^2)
        ----------------
               -2x^2 - 5

Step 5: Bring Down the Next Term

Bring down the next term from the dividend (-5):

        2x^2 ______
x + 3 | 2x^3 + 4x^2 - 5 
        -(2x^3 + 6x^2)
        ----------------
               -2x^2 - 5

Step 6: Repeat Steps 2-5

The leading term of the new dividend (-2x^2 - 5) is -2x^2.
Divide by the leading term of the divisor (x): -2x^2 / x = -2x. Write this above the dividend.

        2x^2 - 2x _____
x + 3 | 2x^3 + 4x^2 - 5 
        -(2x^3 + 6x^2)
        ----------------
               -2x^2 - 5

Multiply the quotient (-2x) by the divisor (x + 3): -2x (x + 3) = -2x^2 - 6x.
Subtract:

        2x^2 - 2x _____
x + 3 | 2x^3 + 4x^2 - 5 
        -(2x^3 + 6x^2)
        ----------------
               -2x^2 - 5 
               -(-2x^2 - 6x)
               ----------------
                       6x - 5

Bring down the next term (there is none).

Step 7: Repeat Again

The leading term of the new dividend (6x - 5) is 6x.
Divide by the leading term of the divisor (x): 6x / x = 6. Write this above the dividend.

        2x^2 - 2x + 6  
x + 3 | 2x^3 + 4x^2 - 5 
        -(2x^3 + 6x^2)
        ----------------
               -2x^2 - 5 
               -(-2x^2 - 6x)
               ----------------
                       6x - 5

Multiply the quotient (6) by the divisor (x + 3): 6 (x + 3) = 6x + 18.
Subtract:

        2x^2 - 2x + 6  
x + 3 | 2x^3 + 4x^2 - 5 
        -(2x^3 + 6x^2)
        ----------------
               -2x^2 - 5 
               -(-2x^2 - 6x)
               ----------------
                       6x - 5 
                       -(6x + 18)
                       ----------------
                               -23

Step 8: The Result

We have reached a point where the degree of the new dividend (-23) is less than the degree of the divisor (x + 3). Therefore, we stop here.

The result of the polynomial long division is:

2x^3 + 4x^2 - 5 = (x + 3)(2x^2 - 2x + 6) - 23

Or, equivalently:

(2x^3 + 4x^2 - 5) / (x + 3) = 2x^2 - 2x + 6 - 23/(x + 3)