if (x-7) and (x-4) are the factors of px^3 + qx^2 - 5x +84, find the values of p and q.

Answer 1

px^3 + qx^2 - 5x +84 = (x-7)(x-4)(ax-b)
= ax^3 - (11a+b)x^2 + (28a+11b)x - 28b

If the polynomials are identical, then

p = a
q = -(11a+b)
-5 = 28a+11b
84 = -28b
So, b = -3
a = 1
p = 1
q = -(11-3) = -8

and the polynomial is

x^3 - 8x^2 - 5x + 84 = (x-7)(x-4)(x+3)