If f(x) and g(x) are two polynomials such that the polynomial P(x)=fx^3+xgx^3 is divisible by x^2+x+1, then P(1) is equal to ___

Question

If f(x) and g(x) are two polynomials such that the polynomial P(x)=fx^3+xgx^3 is divisible by x^2+x+1, then P(1) is equal to ___ .

MARKS Content Team · Accepted Answer

Correct answer: 0. P(x)=fx^3+xgx^3 P(1)=f(1)+g(1) ...1 Now P(x) is divisible by x^2+x+1 ⇒P(x)=Q(x)x^2+x+1 P(ω)=0=Pω^2 where ω,ω^2 are non-real cube roots of units P(x)=fx^3+xgx^3 P(ω)=fω^3+ωgω^3=0 f(1)+ωg(1)=0 ...2 Pω^2=fω^6+ω^2gω^6=0 f(1)+ω^2g(1)=0 ...3 Now, (2)+(3) ⇒2f(1)+ω+ω^2g(1)=0 2f(1)=g(1) ...4 Now, (2)-(3) ⇒ω-ω^2g(1)=0 g(1)=0=f(1) from (4) from (1), P(1)=f(1)+g(1)=0

If $f (x)$ and $g (x)$ are two polynomials such that the polynomial $P (x) = f (x^{3}) + x g (x^{3})$ is divisible by $x^{2} + x + 1$ , then $P (1)$ is equal to ___ .

Solution

Relevant Questions

If f(x) and g(x) are two polynomials such that the polynomial P(x)=fx3+xgx3 is divisible by x2+x+1, then P(1) is equal to ___ .

Solution

Relevant Questions

If $f (x)$ and $g (x)$ are two polynomials such that the polynomial $P (x) = f (x^{3}) + x g (x^{3})$ is divisible by $x^{2} + x + 1$ , then $P (1)$ is equal to ___ .