Question

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

Answer 1

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

1. **Setup the division**:
We write \( 15x^2 + 14x - 8 \) (the dividend) under the long division symbol and \( 3x + 4 \) (the divisor) outside.

2. **Divide the leading term**:
We start by dividing 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**:
Next, we multiply \( 5x \) by the entire divisor \( 3x + 4 \):

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

Now we subtract \( (15x^2 + 20x) \) from the original polynomial:

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

4. **Bring down the next term**:
Since we only have the remainder \( -6x - 8 \) at this point, we don't need to bring down anything (as there are no remaining terms to bring down).

5. **Repeat the process**:
Now, we divide the new leading term \( -6x \) by \( 3x \):

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

6. **Multiply and subtract again**:
We multiply \( -2 \) by the entire divisor:

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

Now, subtract \( (-6x - 8) \) from the current remainder \( -6x - 8 \):

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

7. **Conclusion**:
There is no remainder, and the division is complete. Our final quotient is:

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