Find the value of c so that ( x − 4 ) is a factor of the polynomial p ( x ), p(x)=cx^3−15x−68

Question

don't know how to solve and set up this equation

Answer 1

By the Factor Theorem, if x-4 is a factor of p(x), then p(4) = 0

 

So, 64c - 60 - 68 = 0

 

                   64c = 128

 

                       c = 2

Answer 2

If something is a factor of something else (polynomial or otherwise), that means when you divide them, the remainder is zero.

 

So in this case, the most direct method is to divide p(x) by (x-4) by either polynomial long division (easier to remember but takes up more space), or synthetic division (harder to remember but more compact).

 

I'll use ~ to separate columns for synthetic division:

 

4]  c~0~15~(-68)

      ~4c~16c~60+64c

     c~4c~15+16c~64c-8

 

In order for the remainder to be zero, that means 64c - 8 = 0. So c = 1/8.