find the value of K such that (x+1) is a factor of x^3-kx^2+x+5
a little synthetic divsion shows that the remainder of the division is 3-k.
We wan the remainder to be zero, so k=3 and we see that
x^3-kx^2+x+5 = (x+1)(x^2-4x+5)
To find the value of K such that (x+1) is a factor of x^3 - kx^2 + x + 5, we need to use the Remainder Theorem.
According to the Remainder Theorem, when a polynomial P(x) is divided by (x - a), the remainder is equal to P(a). In this case, we want to find the value of K such that the remainder is equal to 0 when the polynomial is divided by (x+1).
Therefore, we substitute the value -1 into the polynomial:
P(-1) = (-1)^3 - k(-1)^2 + (-1) + 5
= -1 - k + (-1) + 5
= -2 - k
To make the remainder zero, we need:
-2 - k = 0
Solving this equation, we find:
k = -2
Therefore, the value of K required for (x + 1) to be a factor of x^3 - kx^2 + x + 5 is K = -2.