Final answer:
To find the value of k, substitute x = 1 into each polynomial and solve for k. For p(x) = kx^2 - √2x + 1, the value of k is √2 - 1. For p(x) = kx^2 - 3x + k, the value of k is 3/2.
Step-by-step explanation:
In order for x - 1 to be a factor of p(x), the expression p(x) must be divisible by x - 1. To find the value of k, we can use the Remainder Theorem. We substitute x = 1 into p(x) and set it equal to 0:
1. p(x) = kx^2 - √2x + 1
When x = 1:
p(1) = k(1)^2 - √2(1) + 1 = 0
Simplifying the equation, we have:
k - √2 + 1 = 0
Now, to find the value of k, we can isolate it by rearranging the equation:
k = √2 - 1
So, for p(x) = kx^2 - √2x + 1, the value of k is √2 - 1.
2. p(x) = kx^2 - 3x + k
When x = 1:
p(1) = k(1)^2 - 3(1) + k = 0
Simplifying the equation, we have:
k - 3 + k = 0
Combining like terms, we obtain:
2k - 3 = 0
Now, to find the value of k, we can isolate it by rearranging the equation:
k = 3/2
So, for p(x) = kx^2 - 3x + k, the value of k is 3/2.