Question

Can you show that `p(x)=2x^3 + 7x^2 + kx - k` is the produce of 3 linear factors, 2 of which are identical? Show that `k` can take 3 distinct values

Answer 1

See below.

Explanation:

Define `q(x)=a(x+b)^2(x+c)` and consider

`p(x)=q(x)` or

`p(x)=2x^3 + 7x^2 + kx - k = a(x+b)^2(x+c)`

Grouping coefficients we get

`(2-a)x^3+(7-(2ab+ac))x^2+(k-(a b^2 + 2 a b c))x-(k+ab^2c)=0`

Now solving for `a,b,c,k`

`{(a=2),(2ab+ac=7),(a b^2 + 2 a b c=k),(ab^2c=-k):}`

we obtain

`((a,b,c,k),(2,-7/4,7,-343/8),(2,0,7/2,0),(2,2,-1/2,4))`