Trouble with factorising a polynomial

Question

I'm supposed to show that:

$$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$

and the required answers are: $$ P=5, Q=4, R=-4 $$

I tried to solve this with partial fractions like so:

$$5(x-1)(x+2) = A(x+3) + B(x-2)$$

$\implies$ $A$=4, $B$=-4
$\implies$ $Q$=4, R=-4

But where does $P$=5 come from?

Or should I have first multiplied out the numerator and denominator and then used long division to solve?

Answer 1

$$\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ gives
$$5(x-1)(x+2)=P(x-2)(x+3)+Q(x+3)+R(x-2)........(1)$$
Now put $x=2, x=-3,x=0$ respectively on both sides of $(1)$ to get, $Q=4,R=-4,P=5$ respectively. Hence you can reach the desired result.

Answer 2

You are given the wrong expression. After some calculation I figure out that in order for these two expressions to be equivalent,

$$\frac{5(x-1)(x-2)}{(x-2)(x+3)} = 5 + \frac{4}{(x-2)} + \frac{-4}{(x+3)}$$

instead of having $\frac{5(x-1)(x-2)}{(x-2)(x+3)}$, it should be

$$\frac{5(x-1)(x+2)}{(x-2)(x+3)}$$

Now solve it again it would work well.

My approach to solving it is

$$\frac{5(x-1)(x+2)}{(x-2)(x+3)} = \frac{5x^2+5-10}{(x-2)(x+3)} =\frac{\frac{5x^2+5-10}{(x-2)}}{(x+3)}$$

$$\frac{\frac{5x^2+5-10}{(x-2)}}{(x+3)} = \frac{5x+15+\frac{40}{(x-2)}}{(x+3)}$$

$$\frac{5x+15+\frac{40}{(x-2)}}{(x+3)} = 5+\frac{40}{(x-2)(x+3)}$$

$$5+\frac{40}{(x-2)(x+3)}=5 + \frac{4}{(x-2)} + \frac{-4}{(x+3)}$$

Answer 3

You don't need to do any long division. In the expression

$$y = \frac{N(x)}{D(x)} = \frac{5(x-1)(x+2)}{(x-2)(x+3)}$$

The numerator $N(x) = 5(x-1)(x+2)$ and denominator $D(x) = (x-2)(x+3)$ are polynomials of degree $2$. Since the roots of $D(x)$ are simple, $y$ can be rewritten in the form:

$$y = P(x) + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$

where $P(x)$ is a polynomial of degree $\deg N(x) - \deg D(x) = 2 - 2 = 0$, i.e. a constant. To evaluate the 3 coefficients, you can evaluate both side at 3 different values of $x$: the two roots of $D(x)$ and $\infty$:

$$\begin{align} P &= \lim_{x\to\infty} \frac{N(x)}{D(x)} = 5\\ Q &= \lim_{x\to 2} (x-2)\frac{N(x)}{D(x)} = \lim_{x\to 2}\frac{5(x - 1)(x+2)}{x+3} = \frac{5(2 - 1)(2+2)}{2+3} = 4\\ R &= \lim_{x\to -3}(x+3)\frac{N(x)}{D(x)} = \lim_{x\to-3}\frac{5(x-1)(x+2)}{x-2} = \frac{5(-3-1)(-3+2)}{-3-2} = -4 \end{align}$$