To divide the polynomial [tex]\( 15x^2 + 14x - 8 \)[/tex] by [tex]\( 3x + 4 \)[/tex] using polynomial long division, follow these steps:
1. Setup the Division:
Write the dividend [tex]\( 15x^2 + 14x - 8 \)[/tex] and the divisor [tex]\( 3x + 4 \)[/tex].
2. Divide the Leading Terms:
Divide the leading term of the dividend [tex]\( 15x^2 \)[/tex] by the leading term of the divisor [tex]\( 3x \)[/tex]:
[tex]\[
\frac{15x^2}{3x} = 5x
\][/tex]
Write [tex]\( 5x \)[/tex] as the first term of the quotient.
3. Multiply and Subtract:
Multiply [tex]\( 5x \)[/tex] by the divisor [tex]\( 3x + 4 \)[/tex]:
[tex]\[
5x \cdot (3x + 4) = 15x^2 + 20x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\][/tex]
4. Repeat the Process:
Now, [tex]\( -6x - 8 \)[/tex] becomes the new dividend.
Divide the leading term [tex]\( -6x \)[/tex] by the leading term of the divisor [tex]\( 3x \)[/tex]:
[tex]\[
\frac{-6x}{3x} = -2
\][/tex]
Write [tex]\( -2 \)[/tex] as the next term in the quotient.
5. Multiply and Subtract Again:
Multiply [tex]\( -2 \)[/tex] by the divisor [tex]\( 3x + 4 \)[/tex]:
[tex]\[
-2 \cdot (3x + 4) = -6x - 8
\][/tex]
Subtract this from the current dividend:
[tex]\[
(-6x - 8) - (-6x - 8) = 0
\][/tex]
Since the remainder is zero, the quotient of the division is [tex]\( 5x - 2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ 5x - 2 \][/tex]
