Divide the third-degree polynomial, 8x3+20x2−12x , by the first-degree polynomial, x+3 , using long division. What is the result?

1 answer

To divide the polynomial \( 8x^3 + 20x^2 - 12x \) by \( x + 3 \) using long division, we follow these steps:

  1. Set up the long division: \[ \begin{array}{r|rrrr} & 8x^2 - 4x + 0 \ \hline x + 3 & 8x^3 + 20x^2 - 12x \ \end{array} \]

  2. Divide the leading term:

    • Divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( x \): \[ \frac{8x^3}{x} = 8x^2 \]
  3. Multiply and subtract:

    • Multiply \( 8x^2 \) by \( x + 3 \) and subtract from the original polynomial: \[ (8x^2)(x + 3) = 8x^3 + 24x^2 \]
    • Subtract this from \( 8x^3 + 20x^2 - 12x \): \[ (8x^3 + 20x^2 - 12x) - (8x^3 + 24x^2) = -4x^2 - 12x \]
  4. Repeat the process:

    • Now, divide \( -4x^2 \) by \( x \): \[ \frac{-4x^2}{x} = -4x \]
    • Multiply \( -4x \) by \( x + 3 \): \[ (-4x)(x + 3) = -4x^2 - 12x \]
    • Subtract this from \( -4x^2 - 12x \): \[ (-4x^2 - 12x) - (-4x^2 - 12x) = 0 \]
  5. Combine results:

    • The dividend has been completely divided with no remainder. Thus, the result of the division is:

\[ \text{Result} = 8x^2 - 4x \]

Therefore, the quotient when \( 8x^3 + 20x^2 - 12x \) is divided by \( x + 3 \) is \( \boxed{8x^2 - 4x} \).