To divide the polynomial \( 8x^3 + 20x^2 - 12x \) by \( x + 3 \) using long division, we follow these steps:
-
Set up the long division: \[ \begin{array}{r|rrrr} & 8x^2 - 4x + 0 \ \hline x + 3 & 8x^3 + 20x^2 - 12x \ \end{array} \]
-
Divide the leading term:
- Divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( x \): \[ \frac{8x^3}{x} = 8x^2 \]
-
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 \( 8x^3 + 20x^2 - 12x \): \[ (8x^3 + 20x^2 - 12x) - (8x^3 + 24x^2) = -4x^2 - 12x \]
-
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 \( -4x^2 - 12x \): \[ (-4x^2 - 12x) - (-4x^2 - 12x) = 0 \]
-
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 \( \boxed{8x^2 - 4x} \).