Question

Use long division to divide the polynomial 15x2+14x−8 by 3x+4. What is the quotient?(1 point)
Responses

5x−2+163x+4

5x−2

5x+2−163x+4

5x+2

Answer 1

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

1. **Divide the leading term:**
Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \):
\[
\frac{15x^2}{3x} = 5x
\]
This is the first term of the quotient.

2. **Multiply and subtract:**
Multiply \( 5x \) by the entire divisor \( 3x + 4 \) and subtract from the original polynomial:
\[
5x(3x + 4) = 15x^2 + 20x
\]
Now subtract:
\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]

3. **Repeat the process:**
Now, take the new polynomial \(-6x - 8\) and repeat the process. Divide \(-6x\) by \(3x\):
\[
\frac{-6x}{3x} = -2
\]
Now multiply \(-2\) by the divisor \(3x + 4\):
\[
-2(3x + 4) = -6x - 8
\]
Subtract again:
\[
(-6x - 8) - (-6x - 8) = 0
\]

Since the remainder is zero, we conclude that:

\[
\text{Quotient} = 5x - 2
\]

Based on the responses provided, the correct answer is:

**5x - 2**