ANSWER
P(3) = 0
Step-by-step explanation:
Factor Theorem is a consequence of Remainder Theorem.
Remainder Theorem states that if polynomial f(x) is divided by a binomial (x - a) then the remainder is f(a).
Factor Theorem states that if f(a) = 0, then the binomial (x - a) is a factor of f(x).
We have the polynomial
P(x) = x^5-3x^4+5x^3-15x^2-6x+18P(x)=x
5
−3x
4
+5x
3
−15x
2
−6x+18
To prove that x-3 is a factor of P, we calculate P(3):
P(3) = 3^5-3*3^4+5*3^3-15*3^2-6*3+18P(3)=3
5
−3∗3
4
+5∗3
3
−15∗3
2
−6∗3+18
P(3) = 243-243+135-135-18+18P(3)=243−243+135−135−18+18
P(3) = 0P(3)=0
Thus, x-3 is a factor of P(x)
