Joshua wants to know if x−5 is a factor of the polynomial P(x)=x^3−5x^2−x+5. Joshua calculates P(5) and applies the Factor Theorem to conclude that x−5 is not a factor of P(x).

Is Josh's conclusion that x−5 is not a factor correct? Why or why not?

a. No, the remainder is 0, therefore x−5 is a factor of P(x).

b. Yes, the remainder is 0, therefore x−5 is not a factor of P(x).

c. Yes, the remainder is −240, therefore x−5 is not a factor of P(x).

d. No, the remainder is −240, therefore x−5 is a factor of P(x).

please help I am very confused!

hard to say, since you don't say what the remainder is.

The Factor Theorem states that if the remainder is 0, then (x-5) is a factor of P(x).

To determine if x-5 is a factor of the polynomial P(x) = x^3 - 5x^2 - x + 5, we need to use the Factor Theorem. The Factor Theorem states that if a polynomial P(x) has a factor (x-k), then P(k) = 0.

In this case, Joshua calculated P(5) to determine if P(x) is divisible by (x-5). If P(5) equals zero, it means that x-5 is a factor of P(x).

Let's substitute x = 5 into the polynomial:
P(5) = (5)^3 - 5(5)^2 - (5) + 5
= 125 - 125 - 5 + 5
= 0

Since P(5) equals zero, we can conclude that x-5 is a factor of P(x).

The correct answer is:
a. No, the remainder is 0, therefore x−5 is a factor of P(x).

Joshua's conclusion is incorrect. The remainder being zero means that x-5 is indeed a factor of P(x).