Given that f(x) =x^3-kx^2-2x+1 gives a remainder k when divided by x-k,find the value of k
a little synthetic division reveals that
f(x) = (x-k)(x^2-2) + (1-2k)
so, 1-2k = k
k = 1/3
f(x) = x^3 - 1/3 x^2 - 2x + 1
= (x-1/3)(x^2-2) + 1/3
To find the value of k, we will use the Remainder Theorem.
According to the Remainder Theorem, if f(x) gives a remainder k when divided by x - k, then f(k) should be equal to k.
So, we need to evaluate f(k) and equate it to k.
Let's substitute k into the given function:
f(k) = k^3 - k*k^2 - 2k + 1
= k^3 - k^3 - 2k + 1
= - 2k + 1
Now we need to equate f(k) to k:
-2k + 1 = k
Simplifying the equation:
1 = 3k
Dividing both sides by 3, we get:
k = 1/3
Therefore, the value of k is 1/3.
To find the value of k, we need to use the Remainder Theorem. According to the Remainder Theorem, if we divide f(x) by x - k and the remainder is k, then substituting x = k into f(x) should give us k as a result.
Let's substitute x = k into the equation f(x) = x^3 - kx^2 - 2x + 1:
f(k) = k^3 - k(k)^2 - 2k + 1
= k^3 - k^3 - 2k + 1
= -2k + 1
Now, we need to equate f(k) to k, since the remainder when dividing f(x) by x - k is k:
-2k + 1 = k
To solve this equation for k, we can move 2k to the left side of the equation:
-2k - k = -1
Simplifying the left side of the equation:
-3k = -1
Now, we can solve for k by dividing both sides of the equation by -3:
k = (-1) / (-3)
k = 1 / 3
Therefore, the value of k is 1/3.