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
?

Question

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
?

BlackTom AI · Accepted Answer

The question asks for the values of A, B, and C in the partial fraction decomposition: 2x^2 + 5x + 9 over (4x + 1)(5x^2 + 1) equals A/(4x + 1) + (Bx + C)/(5x^2 + 1).

First, clear the denominators by equating the numerators: 2x^2 + 5x + 9 = A(5x^2 + 1) + (Bx + C)(4x + 1).

Expand the right-hand side:
- A(5x^2 + 1) gives 5A x^2 + A.
- (Bx + C)(4x + 1) gives (Bx)(4x) + (Bx)(1) + C(4x) + C = 4B x^2 + Bx + 4C x + C.
Combine like terms: x^2 terms: 5A + 4B; x terms: B + 4C; constant term: A + C.

Set up the system by matching coefficients with 2x^2 + 5x + 9:
- 5A + 4B = 2  (actually 5A + 4B = 2)
- B + 4C = 5
- A + C = 9

Solve step by step:
From A + C = 9, we get A = 9 − C.
Plug into 5A + 4B = 2: 5(9 − C) + 4B = 2 → 45 − 5C + 4B = 2 → 4B − 5C = −43.
From B + 4C = 5, we get B = 5 − 4C.
Substitute B into 4B − 5C = −43: 4(5 − 4C) − 5C = −43 → 20 − 16C − 5C = −43 → 20 − 21C = −43 → −21C = −63 → C = 3.
Then B = 5 − 4C = 5 − 12 = −7.
And A = 9 − C = 9 − 3 = 6.

Therefore, the values are A = 6, B = −7, and C = 3. Note that the question as provided does not list any answer options to select from, so the above values are the outcomes of the algebraic solving procedure for the given decomposition.

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 ?单项选择题

类似问题

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).

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).

[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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