Question

Considerando os polinomios p(x)=2x^4+8x³+x²-4x+1 e q (x)=x^5+2x^4-8x³+x²-4x+1,deter... para quais valores de x temos p(x)=q(x)?

Answer 1

$P(x)=2x^{4}+8x^{3}+x^{2}-4x+1\\Q(x)=x^{5}+2x^{4}-8x^{3}+x^{2}-4x+1$

Fazendo P(x) = Q(x):

$P(x)=Q(x)\\2x^{4}+8x^{3}+x^{2}-4x+1=x^{5}+2x^{4}-8x^{3}+x^{2}-4x+1\\0+8x^{3}+0+0+0=x^{5}+0-8x^{3}+0+0+0\\8x^{3}=x^{5}-8x^{3}\\0=x^{5}-8x^{3}-8x^{3}\\x^{5}-16x^{3}=0$

Colocando x³ em evidência:

$x^{3}(x^{2}-16)=0$

Igualando ambos a zero:

$x^{3}=0~~~\therefore~~~x=\sqrt[3]{0}~~~\therefore~~~x=0\\x^{2}-16=0~~~\therefore~~~x^{2}=16~~~\therefore~~~x=\pm\sqrt{16}~~~\therefore~~~x=\pm4\\\\\\\boxed{\boxed{S=\{-4,0,4\}}}$