Select the best answer for the question.

What is the quotient of [tex]\frac{b^3 + 4b^2 - 3b + 128}{b + 7}[/tex]?

A. [tex]b^2 - 3b + 18 \text{ R } 252[/tex]
B. [tex]b^2 - 3b + 18[/tex]
C. [tex]b^2 - 11b + 38 \text{ R } 140[/tex]
D. [tex]b^2 + 11b + 80 \text{ R } 276[/tex]



Answer :

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.