Question

1) Dados os polinômios A(x)= x^3 - 2x^2 + 4x - 1, B(x) x^2 - 4x + 2 e C(x) x^2 + 1, determine:

a) P(x) = A(x) - 2B(x) + 3C(x)

b) Q(x) = [ B(x) - C(x)]^2 + 2A(x)

Answer 1

Olá,

dados os polinômios

$A(x)= x^{3}-2 x^{2} +4x-1\\ B(x)= x^{2} -4x+2\\ C(x)= x^{2} +1$

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$P(x)=A(x)-2B(x)+3C(x)\\\\ P(x)= x^{3}-2 x^{2} +4x-1-2( x^{2} -4x+2)+3( x^{2} +1)\\ P(x)= x^{3}-2 x^{2} +4x-1-2 x^{2} +8x-4+3 x^{2} +3\\ P(x)= x^{3}-2 x^{2} -2 x^{2} +3 x^{2} +4x+8x-1-4+3\\\\ \boxed{P(x)= x^{3}-x^{2} +12x-2}$

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$Q(x)=[B(x)-C(x)]^2+2A(x)\\\\ Q(x)=[( x^{2} -4x+2)-( x^{2} +1)]^2+2( x^{3}-2 x^{2} +4x-1)\\ Q(x)=[ x^{2} - x^{2} -4x+2-1]^2+2 x^{3}-4 x^{2} +8x-2\\ Q(x)=(1-4x)^2+2x^{3}-4 x^{2} +8x-2\\ Q(x)=1-8x+16 x^{2} +2 x^{3}-4 x^{2} +8x-2\\ Q(x)=2 x^{3}+16 x^{2} -4 x^{2} -8x+8x+1-2\\\\ \boxed{Q(x)=2 x^{3}+12 x^{2} -1}$

Espero ter ajudado e tenha ótimos estudos =))