remainder theorem

Question

32. Use the Remainder Theorem to determine whether x + 7 is a factor of the polynomial

F(x)=x^4+5x^3+22x^2+180x-504

Answer 1

x + 7 = x - (-7)

By the Remainder Theorem, the remainder when F(x) is divided by x + 7 is F(-7)

If F(-7) = 0, then, by the Factor Theorem, x + 7 is a factor of F(x). Otherwise, x + 7 is not a factor of F(x).

F(-7) = 2401 - 1715 + 1078 - 1260 - 504 = 0. Therefore, by the Remainder and Factor Theorems,

x+7 is a factor of F(x).