Question

Use long division to divide.

15x2+14x−83x+4


(1 point)
Responses

5x−2+163x+4
5 x minus 2 plus the fraction with numerator 16 and denominator 3 x plus 4

5x−2
5 x minus 2

5x+2
5 x plus 2

5x+2−163x+4
5 x plus 2 minus the fraction with numerator 16 and denominator 3 x plus 4

Answer 1

To perform the long division of \( 15x^2 + 14x - 8 \) by \( 3x + 4 \), let's follow the steps of long division:

1. **Divide the leading term:** The leading term of the dividend \( 15x^2 \) divided by the leading term of the divisor \( 3x \) gives \( 5x \).

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

3. **Repeat the process:** Now divide the leading term \( -6x \) by the leading term of the divisor \( 3x \):
\[
-6x \div 3x = -2
\]
Multiply \( -2 \) by \( (3x + 4) \):
\[
-2(3x + 4) = -6x - 8
\]
Subtract that from the previous result:
\[
(-6x - 8) - (-6x - 8) = 0
\]

The remainder is \( 0 \), and we have found that:
\[
\frac{15x^2 + 14x - 8}{3x + 4} = 5x - 2
\]

Thus, the final answer is:
\[
\boxed{5x - 2}
\]