Question

How do you find the value of k such that x − 3 is a factor of `x^3 − kx^2 + 2kx − 15`?

Answer 1

`k=4`

Explanation:

We should attempt synthetic division to find when `(x^3-kx^2+2kx-15)/(x-3)` has a remainder of `0`, which would signify that it is a factor of the polynomial.

The synthetic substitution would be set up as:

`{:(ul3|,1,-k,2k,-15),(ul" ",ul" ",ul" ",ul" ",ul" "),(" "," "," "," ","|"0):}`

Treating the synthetic division like any other synthetic division problem, we see that

`{:(ul3|,1," "-k," "2k," "" "color(red)(-15)),(ul" ",ul" ",ul(" "3" "" "),ul(-3k+9),ul(color(red)(-3k+27))),(" ",1,-k+3,-k+9,"|"0):}`

If the remainder equals `0`, then we know that

`-3k+27+(-15)=0`

Solve this to see that `k=4`.

Thus, `(x-3)` is a factor of `x^3-4x^2+8x-15`.