Question

Use long division to divide the polynomial 15x^2+14x−8 by 3x+4 . What is the quotient?

Answer 1

To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, follow these steps:

1. **Set up the long division**: Write \( 15x^2 + 14x - 8 \) under the division symbol and \( 3x + 4 \) outside.

\[
\begin{array}{r|r}
3x + 4 & 15x^2 + 14x - 8 \\
\end{array}
\]

2. **Divide the leading term**: Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \):

\[
\frac{15x^2}{3x} = 5x
\]

3. **Multiply and subtract**: Multiply \( 5x \) by \( 3x + 4 \) and subtract the result from the original polynomial.

\[
5x(3x + 4) = 15x^2 + 20x
\]

Now subtract:

\[
15x^2 + 14x - 8 - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]

4. **Repeat the process**: Now use \( -6x - 8 \) as the new dividend. Divide the leading term \( -6x \) by \( 3x \):

\[
\frac{-6x}{3x} = -2
\]

5. **Multiply and subtract again**: Multiply \( -2 \) by \( 3x + 4 \) and subtract:

\[
-2(3x + 4) = -6x - 8
\]

Subtract:

\[
-6x - 8 - (-6x - 8) = 0
\]

6. **Final quotient**: Since there is no remainder, we conclude the division here.

The quotient from the long division of \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) is:

\[
\boxed{5x - 2}
\]