Question

Use long division to divide the polynomial 15x2+14x−8 by 3x+4. What is the quotient?(1 point)Responses5x+2−163x+45 x plus 2 minus Start Fraction 16 over 3 x plus 4 End Fraction5x−2+163x+45 x minus 2 plus Start Fraction 16 over 3 x plus 4 End Fraction5x−25 x minus 25x+2

Answer 1

To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, let's go through the steps:

1. **Setup the long division**:
\[
\text{Divide: } 15x^2 + 14x - 8 \div (3x + 4)
\]

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

Subtract this product from the original polynomial:
\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]

4. **Repeat the process**:
Now take the leading term \(-6x\) and divide it by \(3x\):
\[
\frac{-6x}{3x} = -2
\]

5. **Multiply and subtract again**:
Multiply \(-2\) by \(3x + 4\):
\[
-2(3x + 4) = -6x - 8
\]

Subtract this from \(-6x - 8\):
\[
(-6x - 8) - (-6x - 8) = 0
\]

6. **Conclusion**:
The result of the division is:
\[
5x - 2
\]
with a remainder of 0. Thus, the quotient of the division \( 15x^2 + 14x - 8 \div (3x + 4) \) is:
\[
\boxed{5x - 2}
\]