if (x-1) is a factor of the polynomial f(x)= 4x^3-4x^2-x-k, where k is a constant, find the value of k.
If (x-1) is a factor of the polynomial f(x), then we can use synthetic division to check if the remainder is zero.
Using synthetic division:
1 | 4 -4 -1 -k
- 4 0 -1
---------------
4 0 -1 -k-1
Since the remainder is (-k-1), in order for (x-1) to be a factor of f(x), the remainder must be zero. Therefore, we have:
-k - 1 = 0
k = -1
So the value of k is -1.
To check if (x-1) is a factor of the polynomial f(x) = 4x^3 - 4x^2 - x - k, you can use the Factor Theorem. According to the Factor Theorem, if (x - a) is a factor of a polynomial, then the polynomial evaluated at x = a should be equal to zero.
In this case, you need to evaluate the polynomial f(x) at x = 1, since (x - 1) is the given factor:
f(1) = 4(1)^3 - 4(1)^2 - 1 - k
= 4(1) - 4(1) - 1 - k
= 4 - 4 - 1 - k
= -1 - k
Since (x - 1) is a factor of the polynomial, f(1) should equal zero. Therefore:
-1 - k = 0
To solve for k, isolate the variable:
k = -1
So, the value of k is -1.
To find the value of k, we need to use the factor theorem. According to the factor theorem, if (x - a) is a factor of a polynomial f(x), then f(a) = 0.
In this case, we are given that (x - 1) is a factor of the polynomial f(x) = 4x^3 - 4x^2 - x - k. So, we can substitute x = 1 into the polynomial and set it equal to zero:
f(1) = 4(1)^3 - 4(1)^2 - 1 - k
= 4 - 4 - 1 - k
= -1 - k
Since (x - 1) is a factor, f(1) = 0. Therefore, we can set -1 - k = 0 and solve for k:
-1 - k = 0
k = -1
So, the value of k is -1.