use the remainder theorem to find the remainder when f(x) is divided by x-2. Then use the factor theorem to determine whether x-2 is a factor of f(x).
f(x)=4x^3-10x^2+13x-13
To find the remainder when dividing a polynomial by a linear equation, we can use the remainder theorem.
First, let's substitute 2 into the polynomial f(x):
f(2) = 4(2)^3 - 10(2)^2 + 13(2) - 13
= 4(8) - 10(4) + 26 - 13
= 32 - 40 + 26 - 13
= 45 - 40
= 5
Therefore, when f(x) = 4x^3 - 10x^2 + 13x - 13 is divided by x - 2, the remainder is 5.
Now, let's check whether x - 2 is a factor of f(x) using the factor theorem. According to the factor theorem, if f(a) = 0, then x - a is a factor of f(x).
To determine if f(x) has a root at x = 2, we can substitute 2 for x in f(x) and check if it equals zero:
f(2) = 4(2)^3 - 10(2)^2 + 13(2) - 13
= 4(8) - 10(4) + 26 - 13
= 32 - 40 + 26 - 13
= 45 - 40
= 5
Since f(2) = 5 ≠ 0, x - 2 is not a factor of f(x).