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

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial 2x⁴ + 3x³ + 5x - 1 by x² + 3x + 2.

Long Division Method

The most common method for dividing polynomials is long division, which mirrors the process of long division with numbers.

1. Set up the Division

Start by setting up the division problem like traditional long division:

            ________
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1

2. Divide the Leading Terms

Focus on the leading terms of both the divisor (x²) and the dividend (2x⁴). Divide 2x⁴ by x², which gives 2x². Write this result above the dividend, aligned with the x² term:

            2x²     
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1

3. Multiply and Subtract

Multiply the quotient term (2x²) by the entire divisor (x² + 3x + 2):

            2x²     
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1
            -(2x⁴ + 6x³ + 4x²)

Subtract the resulting expression from the dividend. Notice that the leading terms cancel out:

            2x²     
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1
            -(2x⁴ + 6x³ + 4x²)
             ----------------
                  -3x³ - 4x² + 5x

4. Repeat the Process

Bring down the next term of the dividend (-1):

            2x²     
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1
            -(2x⁴ + 6x³ + 4x²)
             ----------------
                  -3x³ - 4x² + 5x - 1

Now, focus on the leading terms of the new dividend (-3x³) and the divisor (x²): Divide -3x³ by x², which gives -3x. Write this term next to the 2x² in the quotient:

            2x² - 3x     
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1
            -(2x⁴ + 6x³ + 4x²)
             ----------------
                  -3x³ - 4x² + 5x - 1

Multiply -3x by the divisor (x² + 3x + 2) and subtract the result:

            2x² - 3x     
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1
            -(2x⁴ + 6x³ + 4x²)
             ----------------
                  -3x³ - 4x² + 5x - 1
                  -(-3x³ - 9x² - 6x)
                  -----------------
                         5x² + 11x - 1

5. Continue Dividing

Repeat the process until the degree of the remaining dividend is less than the degree of the divisor. In this case, we need to divide 5x² by x², giving us 5:

            2x² - 3x + 5   
x² + 3x + 2 | 2x⁴ + 3x³ + 0x² + 5x - 1
            -(2x⁴ + 6x³ + 4x²)
             ----------------
                  -3x³ - 4x² + 5x - 1
                  -(-3x³ - 9x² - 6x)
                  -----------------
                         5x² + 11x - 1
                         -(5x² + 15x + 10)
                         --------------
                                -4x - 11

6. Express the Result

The final result of the division is:

(2x⁴ + 3x³ + 5x - 1) / (x² + 3x + 2) = 2x² - 3x + 5 + (-4x - 11) / (x² + 3x + 2)

This represents the quotient (2x² - 3x + 5) and the remainder (-4x - 11), which is still divided by the original divisor (x² + 3x + 2).