To find the quotient of the polynomial [tex]\(\frac{b^3 + 4b^2 - 3b + 128}{b + 7}\)[/tex], we will perform polynomial long division. Here is the step-by-step solution:
1. Divide the leading term of the numerator by the leading term of the divisor:
The leading term of the numerator is [tex]\(b^3\)[/tex], and the leading term of the divisor is [tex]\(b\)[/tex].
[tex]\[
\frac{b^3}{b} = b^2
\][/tex]
So, [tex]\(b^2\)[/tex] is the first term of the quotient.
2. Multiply the entire divisor by this term:
[tex]\[
b^2 \cdot (b + 7) = b^3 + 7b^2
\][/tex]
3. Subtract this from the original polynomial:
[tex]\[
(b^3 + 4b^2 - 3b + 128) - (b^3 + 7b^2) = (4b^2 - 7b^2) + (-3b) + 128 = -3b^2 - 3b + 128
\][/tex]
4. Repeat the process for the new polynomial [tex]\(-3b^2 - 3b + 128\)[/tex]:
Divide the leading term [tex]\(-3b^2\)[/tex] by [tex]\(b\)[/tex]:
[tex]\[
\frac{-3b^2}{b} = -3b
\][/tex]
So, [tex]\(-3b\)[/tex] is the next term of the quotient.
5. Multiply the entire divisor by this term:
[tex]\[
-3b \cdot (b + 7) = -3b^2 - 21b
\][/tex]
6. Subtract this from the polynomial:
[tex]\[
(-3b^2 - 3b + 128) - (-3b^2 - 21b) = -3b - (-21b) + 128 = 18b + 128
\][/tex]
7. Repeat the process for the new polynomial [tex]\(18b + 128\)[/tex]:
Divide the leading term [tex]\(18b\)[/tex] by [tex]\(b\)[/tex]:
[tex]\[
\frac{18b}{b} = 18
\][/tex]
So, [tex]\(18\)[/tex] is the next term of the quotient.
8. Multiply the entire divisor by this term:
[tex]\[
18 \cdot (b + 7) = 18b + 126
\][/tex]
9. Subtract this from the polynomial:
[tex]\[
(18b + 128) - (18b + 126) = 128 - 126 = 2
\][/tex]
So, the quotient is [tex]\(b^2 - 3b + 18\)[/tex] and the remainder is [tex]\(2\)[/tex].
Therefore, the answer is:
B. [tex]\(b^2 - 3b + 18\)[/tex].
There is no remainder option in the choices, but for reference, the remainder is indeed 2 if you were to fully complete the division. The correct answer for the quotient alone is B.
