Question

How to prove that the expression `x^3+(k-1)x^2+(k-9)x-7` is divisible by (x-1) for all values of k?

Answer 1

`"One cannot prove this as it is not true."`

Explanation:

`"A polynomial f(x) is divisible by (x-a) if and only if f(a) = 0."`
`"So we calculate f(1) here and get 1 + k-1 + k-9 - 7 = 2k-16"`
`"and this is only 0 for k=8."`
`"What is true is that it is divisible by (x+1) for all k because"`
`"f(-1) = -1 + k-1 -k + 9 - 7 = 0 for all values of k."`

Answer 2

`"The poly is divisible by "(x-1) iff k=8`.

Explanation:

A polynomial is divisible by `(x-1) iff` the sum of the co-effs. is zero.

In our case, the sum of the co-effs. is given by,

`1+(k-1)+(k-9)-7=2k-16`.

`:.(2k-16)=0, or, k=8 iff" the poly. is divisible by "(x-1)`.

Answer 3

See below.

Explanation:

`p(x) = x^3 + (k - 1) x^2 + (k - 9) x - 7`

is divisible by `x+1` giving

`p(x) = (x+1)(x^2+(k-2)x-7)`

so the claim should be divisible by `x+1`