Find the value of c so that (x-3) is a factor of the polynomial p(x)= -x^3 + cx^2 - 4x + 3

Answer 1

For a given polynomial P(x), the remainder theorem states that P(a) gives the remainder when P(x) is divided by (x - a). So, for (x - a) to be a factor of P(x), it must divide P(x) completely leaving no remainder. That is, P(a) should be zero. In our case P(x) = - x³ + cx² - 4x + 3 and a = 3. So, x - 3 is a factor of P(x) = - x³ + cx² - 4x + 3 if

P(3) = 0

- (3)³ + c(3)² - 4(3) + 3 = 0

- 27 + 9c - 12 + 3 = 0

9c = 36

c = 4