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.

SMAT011 Weekly Quiz 6 |LA005

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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