Consider the rational function f(x)=(x+1)/( (x+9)
(
𝑥
−
1
)
2
)

It can be rewritten using partial fractions in the form

f(x) = A/(x+9) + B/(x-1) + C/
(
𝑥
−
1
)
2

What is the value of C?

Round your answer to 4 decimal places. If Canvas removes trailing zeros after the rounded digit, then you may use fewer decimal places (e.g., 0.2500 would mean your answer should be entered as 0.25).

Question

Consider the rational function f(x)=(x+1)/( (x+9)
(
𝑥
−
1
)
2
 )

It can be rewritten using partial fractions in the form

f(x) = A/(x+9) + B/(x-1) + C/
(
𝑥
−
1
)
2

What is the value of C?

Round your answer to 4 decimal places. If Canvas removes trailing zeros after the rounded digit, then you may use fewer decimal places (e.g., 0.2500 would mean your answer should be entered as 0.25).

BlackTom AI · Accepted Answer

For partial fractions with a repeated linear factor (x-1)^2, the decomposition must include B/(x-1) and C/(x-1)^2 in addition to A/(x+9). So write

(x+1)/[(x+9)(x-1)^2] = A/(x+9) + B/(x-1) + C/(x-1)^2.

1) Relevant concepts
- Partial fraction decomposition for distinct and repeated linear factors.
- Multiplying through by the full denominator to obtain a polynomial identity and then solving for unknown coefficients (often by substituting convenient x-values).

2) Multiply both sides by (x+9)(x-1)^2 to get an identity:
x + 1 = A(x-1)^2 + B(x+9)(x-1) + C(x+9).

3) Solve for C by choosing x = 1 (this zeroes the terms multiplied by (x-1)):
Left side: 1 + 1 = 2.
Right side: A·0 + B·(1+9)·0 + C·(1+9) = 10C.

Thus 2 = 10C.

4) Solve for C:
C = 2/10 = 0.2.

Why this is correct: substituting x = 1 isolates C because the factors (x-1) and (x-1)^2 annihilate the A and B contributions, leaving only C(x+9). That gives an exact, unique value for C consistent with the polynomial identity. If one expanded and equated coefficients instead, the same value would result.

Why other values would be incorrect: any coefficient other than 0.2 would make the identity fail at x = 1 (for example, C = 0.1 gives right side 1 instead of 2). The algebraic isolation here leaves no ambiguity.

Conclusion: The coefficient C equals 0.2, matching the provided answer.

Similar Questions

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In the partial fraction decomposition 2 𝑥 2 + 5 𝑥 + 9 ( 4 𝑥 + 1 ) ( 5 𝑥 2 + 1 ) = 𝐴 4 𝑥 + 1 + 𝐵 𝑥 + 𝐶 5 𝑥 2 + 1 , what are the values of 𝐴 , 𝐵 and 𝐶 ?

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