Michael Standfest

12th grade

student pre-calc
If x+4 is a factor of 2x^{4}+kx^{3}-3kx^{2}+6x-40, find k

and

Prove that n^{2}-n is even for all n, using the proof of contradiction

Hi Michael,

There is a wonderful mathematical result called The Factor Theorem which implies that if f(x) is a polynomial then x - a is a factor of f(x) if and only if f(a) = 0. Hence x + 4 is a factor of f(x) = 2x^{4}+kx^{3}-3kx^{2}+6x-40 if and only if f(-4) = 0. Thus substitute x = -4 into f(x) = 0 and solve for k.

For the second problem suppose that for some particular n, call it k, k^{2}-k = k(k-1) is odd. Now show that k cannot be even and it cannot be odd. This is the contradiction.