SOLUTION: If P(x)=(g(x^3))+(xh(x^3)) is divisible by (x^2)+x+1,then show that g(x) and h(x) are divisible by (x-1). Thanks

Question 917729: If P(x)=(g(x^3))+(xh(x^3)) is divisible by (x^2)+x+1,then show that g(x) and h(x) are divisible by (x-1).

Thanks
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

for some quotient polynomial and some number remainder .
(If , then is divisible ).
for some quotient polynomial and some number remainder .
(If , then is divisible ).

Substituting, we get

Since ,

That means that
when you divide by the quadratic polynomial ,
the quotient is ,
and the remainder is the linear polynomial .
Since is divisible by ,
must be zero for all values of , meaning that .
RELATED QUESTIONS
If g(x)=x^2+3x-1,find (g(x+h)-g(x))/h The answer is 2x+hbut I keep getting 2x+h+3... (answered by ewatrrr)

Part 1: Let f(x) and g(x) be polynomials. Suppose f(x)=0 for exactly three values... (answered by ikleyn)

determine if p(x) is divisible by the given binomial 1.p(x)= x^5+32; x+2 2.p(x)=... (answered by ikleyn)

if g(x) = x^2 + 2 and h(x) = x - 3, what is... (answered by Tatiana_Stebko,solver91311)

Part 1: Let $f(x)$ and $g(x)$ be polynomials. Suppose $f(x)=0$ for exactly three... (answered by greenestamps,Edwin McCravy)

If an integer x is divisible by 7, then x^2 is divisible by... (answered by tommyt3rd)

If -2 and 4 are both zeros of the polynomial f(x), then a factor of f(x) is divisible by (answered by ikleyn)

If f(x)=x�+3, and g(x)=x-1 then f(g(x))... (answered by nyc_function)