4. (a) Find the values of the constants A, B and C - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 7

To find constants A, B, and C, we can use partial fraction decomposition on the expression:

$\frac{2(4x^2 + 1)}{(2x + 1)(2x - 1)} = \frac{A}{(2x + 1)} + \frac{B}{(2x - 1)} + \frac{C}{(2x + 1)(2x - 1)}$

By multiplying both sides by the denominator ((2x + 1)(2x - 1)), we have:

$2(4x^2 + 1) = A(2x - 1) + B(2x + 1) + C$

Expanding the right-hand side gives:

$2(4x^2 + 1) = (2Ax - A) + (2Bx + B) + C$

Now, regrouping terms:

$2(4x^2 + 1) = (2A + 2B)x + (-A + B + C)$

Matching coefficients:

• For $x^2$: No contribution, so this doesn't give useful information.
• For $x$: $2A + 2B = 0$, leading to $A + B = 0$. Thus, $B = -A$.
• For the constant: $-A + B + C = 2$.

Substituting $B = -A$, we get: $-A - A + C = 2 \Rightarrow C = 2 + 2A$

We can select $A = 2$, so:

• $A = 2$
• $B = -2$
• $C = 6$.