Find the values of the constants A, B and C - Edexcel - A-Level Maths Pure - Question 7 - 2010 - Paper 6
Question 7
Find the values of the constants A, B and C.
Hence, or otherwise, expand \( \frac{2x^2 + 5x - 10}{(x - 1)(x + 2)} \) in ascending powers of x, as far as the term in... show full transcript
Worked Solution & Example Answer:Find the values of the constants A, B and C - Edexcel - A-Level Maths Pure - Question 7 - 2010 - Paper 6
Step 1
Find the values of the constants A, B and C.
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Answer
To find the constants A, B, and C, we start by performing partial fraction decomposition:
(x−1)(x+2)2x2+5x−10≡x−1A+x+2B+x+2C
Multiply both sides by ((x - 1)(x + 2)
2x2+5x−10=A(x+2)+B(x−1)(x+2)+C(x−1)
Set up equations by substituting suitable values to eliminate A, B, and C.
Let ( x = 1 ):
2(1)2+5(1)−10=A(1+2)2+5−10=3A⇒−3=3A⇒A=−1
Let ( x = -2 ):
2(−2)2+5(−2)−10=C(−2−1)8−10−10=−3C⇒−12=−3C⇒C=4
To find B, we can use a different value. Let's choose ( x = 0 ):
2(0)2+5(0)−10=A(0+2)+B(−1)(2)+C(−1)−10=A(2)+B(−2)+C(−1)
Plugging in the values of A and C:
−10=−1(2)+B(−2)+4(−1)−10=−2−2B−4−10=−6−2B⇒−4=−2B⇒B=2
Thus, we have found:
A = 2
B = -1
C = 4.
Step 2
Hence, or otherwise, expand \( \frac{2x^2 + 5x - 10}{(x - 1)(x + 2)} \) in ascending powers of x, as far as the term in \( x^2 \).
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Answer
Now using the constants found, we can express:
(x−1)(x+2)2x2+5x−10=x−12+x+2−1+x+24
This becomes:
Combine like terms:
( \frac{2}{x-1} + \frac{3}{x + 2} )
Perform polynomial long division on each term:
For ( \frac{2}{x - 1} ), expand:
x−12=−2(x1(1−x1)−1)
For ( x = 0 ):
This gives the final expression:
2x2+... with ( A - B + C = 5 ) thereby giving the correct expansion up to the term in ( x^2 ).\n - Hence, we find:
( \frac{5}{2} \cdots x^{2} + ... ).
