C3 January 2008 Q1
1. Given that\[\frac{2x^4 - 3x^2 + x + 1}{(x^2 - 1)} \equiv (ax^2 + bx + c) + \frac{dx + e}{(x^2 - 1)},\]find the values of the constants \(a\), \(b\), \(c\), \(d\) and \(e\). (4)
| Scheme | Marks |
|---|---|
| \[\begin{array}{rl} & \ \ \ 2x^2 \qquad\ \ -1 \\ x^2 - 1 & \big)\overline{\,2x^4 - 3x^2 + x + 1} \\ & \ \ \ \underline{2x^4 - 2x^2\phantom{{}+x+1}} \\ & \ \ \ \phantom{2x^4}\ \ \, -x^2 + x + 1 \\ & \ \ \ \phantom{2x^4}\ \ \, \underline{-x^2 \phantom{{}+x} + 1} \\ & \ \ \ \phantom{2x^4 -x^2}\ \ \ \ \ \ x \end{array}\] | M1 |
| \(a = 2\) stated or implied | A1 |
| \(c = -1\) stated or implied | A1 |
| \(2x^2 - 1 + \dfrac{x}{x^2 - 1}\) | |
| \(a = 2,\ b = 0,\ c = -1,\ d = 1,\ e = 0\) \(d = 1\) and \(b = 0,\ e = 0\) stated or implied | A1 |
| (4 marks) |