P (x)=x3 +kx -2x +1, when p (x) is divided by (x-k), remainder is k, find the all the values for k. Use the remainder theorem

Question

This is from Algebra, but i already have done, by using other opyions, but using a remainder theorem is very hard

Answer 1

By the Remainder Theorem, P(k) is the remainder when P(x) is divided by        x-k.  

 

So, P(k) = k

 

k³ + k² - 2k + 1 = k

 

k³ + k² - 3k + 1 = 0

 

By inspection, we see that 1 is a root.

 

Divide synthetically by k - 1:

 

1⌋    1      1       -3     1

    ______1___       2__-1

        1      2       -1    0

 

So (k-1)(k² + 2k - 1) = 0

 

If k² + 2k - 1 = 0, then k = -2 ± √8 = -1± √2

                                                   2

 

k = 1, -1 + √2, -1 - √2