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) = ${\displaystyle \frac{p\left(x\right)-q\left(x\right)}{6}}$(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