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?

3 Answers
Jan 24, 2018

#"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."#

Jan 24, 2018

#"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)#.

Jan 24, 2018

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#