Consider the rational function f(x)=(x+1)/( (x+3)(6+
𝑥
2
) )

It can be rewritten using partial fractions in the form

f(x) = A/(x+3) + (Bx+C)/(6+
𝑥
2
)

What is the value of B?

Use a maximum of 4 decimal places to enter your answer (i.e. if your answer requires more than 4DP, round it to 4DP).

Question

Consider the rational function f(x)=(x+1)/( (x+3)(6+
𝑥
2
) )

It can be rewritten using partial fractions in the form

f(x) = A/(x+3) + (Bx+C)/(6+
𝑥
2
 )

What is the value of B?

Use a maximum of 4 decimal places to enter your answer (i.e. if your answer requires more than 4DP, round it to 4DP).

BlackTom AI · Accepted Answer

Write (x+1)/((x+3)(x^2+6)) in the given partial-fraction form:
(x+1)/((x+3)(x^2+6)) = A/(x+3) + (Bx+C)/(x^2+6).

Multiply both sides by (x+3)(x^2+6) to eliminate denominators:
x + 1 = A(x^2 + 6) + (Bx + C)(x + 3).

Expand the right-hand side and collect like terms:
A(x^2 + 6) = A x^2 + 6A.
(Bx + C)(x + 3) = B x^2 + 3B x + C x + 3C = B x^2 + (3B + C)x + 3C.

Add terms:
x + 1 = (A + B)x^2 + (3B + C)x + (6A + 3C).

Equate coefficients with the left-hand side (which is 0·x^2 + 1·x + 1):
Coefficient of x^2: A + B = 0
Coefficient of x:   3B + C = 1
Constant term:      6A + 3C = 1

Solve the system. From A + B = 0, A = −B.
Substitute A into the constant-equation: 6(−B) + 3C = 1 → −6B + 3C = 1 → 3C = 1 + 6B → C = (1 + 6B)/3.
Substitute C into the x-coefficient equation: 3B + (1 + 6B)/3 = 1.
Multiply by 3 to clear denominators: 9B + 1 + 6B = 3 → 15B + 1 = 3 → 15B = 2 → B = 2/15.

Convert to decimal and round to four decimal places: 2/15 = 0.133333... → 0.1333.

Verify quickly: A = −B = −2/15, C = (1 + 6B)/3 = (1 + 12/15)/3 = 3/5; substituting gives the original numerator coefficients, confirming correctness.

Any other rounded value (for example 0.1332 or 0.1334) would be incorrect due to rounding rules. Therefore the value of B is 0.1333, which matches the provided answer.

类似问题

Consider the rational function f(x)=(x+1)/( (x+4)(5+ 𝑥 2 ) ) It can be rewritten using partial fractions in the form f(x) = A/(x+4) + (Bx+C)/(5+ 𝑥 2 ) What is the value of B? Use a maximum of 4 decimal places to enter your answer (i.e. if your answer requires more than 4DP, round it to 4DP).

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 ?

[math: 4x+1(x+4)2(x2+4)] \frac{4x+1}{(x+4)^2(x^2+4)} can be expressed in partial fractions as:

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