Answer :
To solve the given problem, we need to perform polynomial division on [tex]\(\frac{b^3 + 4b^2 - 3b + 120}{b + 7}\)[/tex].
### Step-by-Step Polynomial Division
Let's denote the dividend by [tex]\( P(b) = b^3 + 4b^2 - 3b + 120 \)[/tex] and the divisor by [tex]\( D(b) = b + 7 \)[/tex].
1. First term of the quotient: To find the first term of the quotient, divide the leading term of [tex]\( P(b) \)[/tex] by the leading term of [tex]\( D(b) \)[/tex]:
[tex]\[ \frac{b^3}{b} = b^2 \][/tex]
So the first term of the quotient is [tex]\( b^2 \)[/tex].
2. Multiply and subtract: Multiply [tex]\( b^2 \)[/tex] by [tex]\( D(b) \)[/tex] and subtract the result from [tex]\( P(b) \)[/tex]:
[tex]\[ (b^3 + 4b^2 - 3b + 120) - (b^2 \cdot (b + 7)) = (b^3 + 4b^2 - 3b + 120) - (b^3 + 7b^2) \][/tex]
Simplifying this gives:
[tex]\[ b^3 + 4b^2 - 3b + 120 - b^3 - 7b^2 = -3b^2 - 3b + 120 \][/tex]
3. Second term of the quotient: Divide the new leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{-3b^2}{b} = -3b \][/tex]
So the second term of the quotient is [tex]\(-3b\)[/tex].
4. Multiply and subtract: Multiply [tex]\(-3b\)[/tex] by [tex]\( D(b) \)[/tex] and subtract the result from the dividend obtained in the last step:
[tex]\[ (-3b^2 - 3b + 120) - (-3b \cdot (b + 7)) = (-3b^2 - 3b + 120) - (-3b^2 - 21b) \][/tex]
Simplifying this gives:
[tex]\[ -3b^2 - 3b + 120 + 3b^2 + 21b = 18b + 120 \][/tex]
5. Third term of the quotient: Divide the new leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{18b}{b} = 18 \][/tex]
So the third term of the quotient is [tex]\( 18 \)[/tex].
6. Multiply and subtract: Multiply [tex]\( 18 \)[/tex] by [tex]\( D(b) \)[/tex] and subtract the result from the dividend obtained in the last step:
[tex]\[ (18b + 120) - (18 \cdot (b + 7)) = (18b + 120) - (18b + 126) \][/tex]
Simplifying this gives:
[tex]\[ 18b + 120 - 18b - 126 = -6 \][/tex]
### Conclusion
The quotient of the polynomial division [tex]\( \frac{b^3 + 4b^2 - 3b + 120}{b + 7} \)[/tex] is [tex]\( b^2 - 3b + 18 \)[/tex] with a remainder of [tex]\( -6 \)[/tex].
Let's review the given answer choices:
A. [tex]\( b^2 - 3b + 18 \)[/tex] R252
B. [tex]\( b^2 - 3b + 18 \)[/tex]
C. [tex]\( b^2 - 11b + 38 \)[/tex] R140
D. [tex]\( b^2 + 11b + 80 \)[/tex] R 276
Given the quotient [tex]\( b^2 - 3b + 18 \)[/tex] and the remainder [tex]\( -6 \)[/tex], none of the provided answer choices are correct because the quotient matches answer A but the remainder does not match any of the provided remainders.
Thus, the best answer is: None of the options.
### Step-by-Step Polynomial Division
Let's denote the dividend by [tex]\( P(b) = b^3 + 4b^2 - 3b + 120 \)[/tex] and the divisor by [tex]\( D(b) = b + 7 \)[/tex].
1. First term of the quotient: To find the first term of the quotient, divide the leading term of [tex]\( P(b) \)[/tex] by the leading term of [tex]\( D(b) \)[/tex]:
[tex]\[ \frac{b^3}{b} = b^2 \][/tex]
So the first term of the quotient is [tex]\( b^2 \)[/tex].
2. Multiply and subtract: Multiply [tex]\( b^2 \)[/tex] by [tex]\( D(b) \)[/tex] and subtract the result from [tex]\( P(b) \)[/tex]:
[tex]\[ (b^3 + 4b^2 - 3b + 120) - (b^2 \cdot (b + 7)) = (b^3 + 4b^2 - 3b + 120) - (b^3 + 7b^2) \][/tex]
Simplifying this gives:
[tex]\[ b^3 + 4b^2 - 3b + 120 - b^3 - 7b^2 = -3b^2 - 3b + 120 \][/tex]
3. Second term of the quotient: Divide the new leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{-3b^2}{b} = -3b \][/tex]
So the second term of the quotient is [tex]\(-3b\)[/tex].
4. Multiply and subtract: Multiply [tex]\(-3b\)[/tex] by [tex]\( D(b) \)[/tex] and subtract the result from the dividend obtained in the last step:
[tex]\[ (-3b^2 - 3b + 120) - (-3b \cdot (b + 7)) = (-3b^2 - 3b + 120) - (-3b^2 - 21b) \][/tex]
Simplifying this gives:
[tex]\[ -3b^2 - 3b + 120 + 3b^2 + 21b = 18b + 120 \][/tex]
5. Third term of the quotient: Divide the new leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{18b}{b} = 18 \][/tex]
So the third term of the quotient is [tex]\( 18 \)[/tex].
6. Multiply and subtract: Multiply [tex]\( 18 \)[/tex] by [tex]\( D(b) \)[/tex] and subtract the result from the dividend obtained in the last step:
[tex]\[ (18b + 120) - (18 \cdot (b + 7)) = (18b + 120) - (18b + 126) \][/tex]
Simplifying this gives:
[tex]\[ 18b + 120 - 18b - 126 = -6 \][/tex]
### Conclusion
The quotient of the polynomial division [tex]\( \frac{b^3 + 4b^2 - 3b + 120}{b + 7} \)[/tex] is [tex]\( b^2 - 3b + 18 \)[/tex] with a remainder of [tex]\( -6 \)[/tex].
Let's review the given answer choices:
A. [tex]\( b^2 - 3b + 18 \)[/tex] R252
B. [tex]\( b^2 - 3b + 18 \)[/tex]
C. [tex]\( b^2 - 11b + 38 \)[/tex] R140
D. [tex]\( b^2 + 11b + 80 \)[/tex] R 276
Given the quotient [tex]\( b^2 - 3b + 18 \)[/tex] and the remainder [tex]\( -6 \)[/tex], none of the provided answer choices are correct because the quotient matches answer A but the remainder does not match any of the provided remainders.
Thus, the best answer is: None of the options.
