# Problem Study (Polynomial)

When a polynomial f(x) is divided by (x - 1) and (x + 5),
the remainders are -6 and 6 respectively. Let r(x be the
remainder when f(x) is divided by x2 + 4x - 5. Find the
value of r(-2).

### Solution

By the problem,
f(x) = p(x) (x - 1) - 6 ............(1)
f(x) = q(x) (x + 5) + 6 ............(2)
f(x) = Q(x) (x2 + 4x - 5) + r(x) ............(3)
(1) × (x + 5) ⇒ (x + 5) f(x) = p(x) (x - 1) (x + 5) - 6x - 30
(2) × (x - 1)  ⇒  (x - 1) f(x) = q(x) (x - 1) (x + 5) + 6x - 6
Subtracting the two equations,
6 f(x) = [p(x) - q(x)] (x - 1) (x + 5) - 12x - 24
Hence ,
f(x) = $\frac{p\left(x\right)-q\left(x\right)}{6}$(x - 1) (x + 5) - 2x - 4 -------(4)
Comparing equations (3) and (4), we have
r(x) = - 2x - 4
Therefore, r (-2) = -2 (-2) - 4 = 0.

Credit : Sayar U Pyi Kyaw