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

3 min read Jun 17, 2024
(x^3+5x^2+5x-2)/(x+2)

Polynomial Long Division: (x³ + 5x² + 5x - 2) ÷ (x + 2)

This article will walk through the process of dividing the polynomial (x³ + 5x² + 5x - 2) by (x + 2) using polynomial long division.

Setting up the Division

  1. Write the dividend and divisor in a long division format.

        ____________
x + 2 | x³ + 5x² + 5x - 2

Performing the Division

  1. Divide the leading term of the dividend (x³) by the leading term of the divisor (x). This gives us x².

        x²_________
x + 2 | x³ + 5x² + 5x - 2

  2. Multiply the quotient (x²) by the divisor (x + 2). This gives us x³ + 2x².

        x²_________
x + 2 | x³ + 5x² + 5x - 2
       x³ + 2x²

  3. Subtract the result from the dividend.

        x²_________
x + 2 | x³ + 5x² + 5x - 2
       x³ + 2x²
       ---------
             3x²

  4. Bring down the next term of the dividend (5x).

        x²_________
x + 2 | x³ + 5x² + 5x - 2
       x³ + 2x²
       ---------
             3x² + 5x

  5. Repeat steps 1-4.

    • Divide the leading term of the new dividend (3x²) by the leading term of the divisor (x). This gives us 3x.
    • Multiply the quotient (3x) by the divisor (x + 2). This gives us 3x² + 6x.
    • Subtract the result.
    • Bring down the next term (-2).
        x² + 3x______
x + 2 | x³ + 5x² + 5x - 2
       x³ + 2x²
       ---------
             3x² + 5x
             3x² + 6x
             ---------
                  -x - 2

  6. Repeat steps 1-4 again.

    • Divide the leading term of the new dividend (-x) by the leading term of the divisor (x). This gives us -1.
    • Multiply the quotient (-1) by the divisor (x + 2). This gives us -x - 2.
    • Subtract the result.
        x² + 3x - 1_
x + 2 | x³ + 5x² + 5x - 2
       x³ + 2x²
       ---------
             3x² + 5x
             3x² + 6x
             ---------
                  -x - 2
                  -x - 2
                  -------
                     0

The Result

We have reached a remainder of zero, which means the division is complete. Therefore, (x³ + 5x² + 5x - 2) ÷ (x + 2) = x² + 3x - 1.

In other words, (x³ + 5x² + 5x - 2) can be factored as (x + 2)(x² + 3x - 1).

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