Express in partial fractions $$\frac{5x + 3}{(2x + 1)(x + 1)^2}$$ - Edexcel - A-Level Maths Pure - Question 3 - 2013 - Paper 1

Multiply both sides by the common denominator, ((2x + 1)(x + 1)^2): $5x + 3 = A(x + 1)^2 + B(2x + 1)(x + 1) + C(2x + 1)$.

Expanding each term:

• For A: $A(x + 1)^2 = A(x^2 + 2x + 1) = Ax^2 + 2Ax + A$
• For B: $B(2x + 1)(x + 1) = B(2x^2 + 3x + 1)$
• For C: $C(2x + 1) = 2Cx + C$ Now combining all terms gives: $5x + 3 = (A + 2B)x^2 + (2A + 3B + 2C)x + (A + B + C)$.

Setting coefficients equal gives us the system of equations:

1. For $x^2$: $A + 2B = 0$
2. For $x$: $2A + 3B + 2C = 5$
3. For the constant: $A + B + C = 3$.

From the first equation, we get: $A = -2B$. Substituting into the other equations:

1. Substituting into equation 3: $-2B + B + C = 3 \implies -B + C = 3 \implies C = B + 3$
2. Now substituting into equation 2: $2(-2B) + 3B + 2(B + 3) = 5$ Simplifying this leads to: $-4B + 3B + 2B + 6 = 5 \implies B + 6 = 5 \implies B = -1$ Using this to find A and C: $A = -2(-1) = 2, \quad C = -1 + 3 = 2$.

Thus, the values are: $A = 2, \quad B = -1, \quad C = 2$. So the partial fraction decomposition is: $\frac{5x + 3}{(2x + 1)(x + 1)^2} = \frac{2}{2x + 1} + \frac{-1}{x + 1} + \frac{2}{(x + 1)^2}$.