2020 WAEC Further Mathematics Theory If \(\frac{3x^2 + 3x - 2}{(x - 1)(x + 1)}\) = P + \(\frac{Q}{x -...

If \(\frac{3x^2 + 3x - 2}{(x - 1)(x + 1)}\) = P + \(\frac{Q}{x - 1} + \frac{R}{x - 1}\)

Find the value of Q and R

\(\frac{3x^2 + 3x - 2}{(x - 1)(x + 1)}\) = P + \(\frac{Q}{x - 1} + \frac{R}{x - 1}\)

Long division

\(x^2 - 1\)

3x\(^2\) + 3x - 2

-(3x\(^2\) - 3x)

\(\overline{6x - 2}\)

\(\frac{6x^{-2}}{(x - 1)(x + 1)}\)

= \(\frac{Q}{x - 1} + \frac{R}{x + 1}\)

6x - 2 = Q (x + 1) + R(x - 1)

Let x = -1

6(-1) - 2 = Q(-1 + 1) + R (-1 - 1)

-6 - 2 = -2R

\(\frac{-8}{-2} = \frac{-2R}{-2}\)

R = 4

From 6x - 2 = Q(x - 1) + R(x - 1)

Let x = 1

6(1) - 2 = Q(1 + 1) + R(1 - 1)

6 - 2 = 2Q

\(\frac{4}{2} = \frac{2Q}{2}\)

= Q = 2

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