1)let f(x) be a polynomial function explian how to use the factor theorem to check if (x-c) is a factor of F(x)

Question

2) use synthetic division to factor X^2-2x^2-9x+18 completly

Answer 1

if f(c) = 0 , then x-c is a factor

for f(x) = x^2 - 2x^2 - 9x + 18
why do you have two x^2 terms, I will assume the first is x^3

If so, we don't need the factor theorem for this one, grouping is obvious

x^3 - 2x^2 - 9x + 18
= x^2(x-2) - 9(x-2)
= (x-2)(x^2 - 9)
= (x-2)(x+3)(x-3)

if you had not seen this, try x = ±1, ±2 , ±3 , that is factors of 18
f(1) ≠ 0
f(-1) ≠ 0
f(2) = 0 , yeahhhh, so x-2 is a factor
..
f(3) = 0 , so x-3 is a factor
f(-3) = 0 , so x+3 is a factor

since we have a cubic, there can only be a maximum of 3 algebraic factors
so as above
(x-2)(x+3)(x-3)