(x ^ 3 + 10x ^ 2 + 13x + 39)/(x ^ 2 + 2x + 1) Rewrite the expression in the form q(x) + r(x). b(x)
A (x + 8) + (- 4x + 31)/(x ^ 2 + 2x + 1)
B (- 4x + 31) + (x + 8)/(x ^ 2 + 2x + 1)
C (x + 8) + (- 4x + 31)/(x ^ 3 + 10x ^ 2 + 13x + 39)
D (- 4x + 31) + (x + 8)/(x ^ 3 + 10x ^ 2 + 13x + 39)
The Correct Answer and Explanation is :
The correct answer is: A (x + 8) + (- 4x + 31)/(x ^ 2 + 2x + 1)
To rewrite the expression ((x^3 + 10x^2 + 13x + 39)/(x^2 + 2x + 1)) in the form (q(x) + \frac{r(x)}{b(x)}), we need to perform polynomial long division.
- Long Division Process:
- Divide the leading term of the numerator ((x^3)) by the leading term of the denominator ((x^2)):
[
x^3 \div x^2 = x
] - Multiply the entire denominator (x^2 + 2x + 1) by (x):
[
x(x^2 + 2x + 1) = x^3 + 2x^2 + x
] - Subtract this result from the original numerator:
[
(x^3 + 10x^2 + 13x + 39) – (x^3 + 2x^2 + x) = 8x^2 + 12x + 39
]
- Repeat the Division:
- Divide (8x^2) by (x^2):
[
8x^2 \div x^2 = 8
] - Multiply the entire denominator (x^2 + 2x + 1) by (8):
[
8(x^2 + 2x + 1) = 8x^2 + 16x + 8
] - Subtract this from the current remainder:
[
(8x^2 + 12x + 39) – (8x^2 + 16x + 8) = -4x + 31
]
- Conclusion:
- The quotient (q(x)) is (x + 8).
- The remainder (r(x)) is (-4x + 31).
- The divisor (b(x)) is (x^2 + 2x + 1).
So, we can rewrite the expression as:
[
\frac{x^3 + 10x^2 + 13x + 39}{x^2 + 2x + 1} = (x + 8) + \frac{-4x + 31}{x^2 + 2x + 1}
]
The correct answer is A: ((x + 8) + \frac{-4x + 31}{x^2 + 2x + 1}).
Explanation
The process of polynomial long division is analogous to numerical long division. We repeatedly divide the leading terms, multiply, and subtract until we reach a degree in the remainder that is lower than that of the divisor. In this case, we derived (q(x)) as (x + 8) and (r(x)) as (-4x + 31), which leads to the final expression in the required form. This method is crucial for simplifying rational functions and is foundational in algebra, as it allows us to express complex polynomials in a more manageable format.