jmc

Question

Let f(x) and g(x) be two monic cubic polynomials, and let r be a real number. Two of the roots of f(x) are r + 1 and r + 7. Two of the roots of g(x) are r + 3 and r + 9, and f(x) - g(x) = rfor all real numbers x. Find r.

Accepted Answer

By Factor Theorem, f(x) = (x - r - 1)(x - r - 7)(x - a)and g(x) = (x - r - 3)(x - r - 9)(x - b)for some real numbers a and b. Then f(x) - g(x) = (x - r - 1)(x - r - 7)(x - a) - (x - r - 3)(x - r - 9)(x - b) = rfor all x. Setting x = r + 3, we get (2)(-4)(r + 3 - a) = r.Setting x = r + 9, we get (8)(2)(r + 9 - a) = r.Then -8r - 24 + 8a = r and 16r + 144 - 16a = r, so align* 8a - 9r &= 24, -16a + 15r &= -144. align*Solving, we find r = 32.