if x-1 is a factor of x^3-2kx^2+3x+1 determine k
by the Remainder Theorem, f(1) = 5-2k
you want a remainder of zero
how did u get this?
To determine the value of k, we need to find the remainder when the polynomial \(x^3 - 2kx^2 + 3x + 1\) is divided by \(x - 1\).
Using the remainder theorem, if \(x - 1\) is a factor of \(x^3 - 2kx^2 + 3x + 1\), then the polynomial evaluated at \(x = 1\) should give us a remainder of 0.
So, let's substitute \(x = 1\) into the polynomial:
\((1)^3 - 2k(1)^2 + 3(1) + 1 = 1 - 2k + 3 + 1 = 5 - 2k\)
Since the remainder is 0, we have:
\(5 - 2k = 0\)
Now, let's solve for k:
\(2k = 5\)
\(k = \frac{5}{2}\)
Therefore, \(k = \frac{5}{2}\) when \(x - 1\) is a factor of \(x^3 - 2kx^2 + 3x + 1\).
To determine the value of k, we need to use the factor theorem. According to the theorem, if x - 1 is a factor of the polynomial x^3 - 2kx^2 + 3x + 1, then substituting x = 1 into the polynomial should yield a result of 0.
Let's substitute x = 1 into the polynomial and solve for k:
x^3 - 2kx^2 + 3x + 1 = 0
(1)^3 - 2k(1)^2 + 3(1) + 1 = 0
1 - 2k + 3 + 1 = 0
5 - 2k = 0
-2k = -5
k = -5 / -2
k = 5/2
Therefore, k is equal to 5/2.