Use long division to divide the polynomial [tex]$15x^2 + 14x - 8$[/tex] by [tex]$3x + 4$[/tex]. What is the quotient?

A. [tex][tex]$5x - 2$[/tex][/tex]

B. [tex]$5x - 2 + \frac{16}{3x + 4}$[/tex]

C. [tex]$5x + 2 - \frac{10}{3x + 4}$[/tex]

D. [tex][tex]$5x + 2$[/tex][/tex]



Answer :

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]