Question

obtenha um polinomio p(x) de grau

a) 3, cujas raízes são
-3, 2 e 3

b) 4, cujas raízes são 3+i, 3-i, 2 e -1

c) 5, cujas raízes são
-3, 1, 0, 1 e 3

Answer 1

Seja P(x) um polinômio de grau 'n':

$P(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+...+a_{n}$

Esse polinômio pode ser escrito em função de suas raízes (α):

$P(x)=a(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3})(x-\alpha_{4})...(x-\alpha_{n})$
_________________________

a)

$P(x)=(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3})\\P(x)=(x-[-3])(x-2)(x-3)\\P(x)=(x+3)(x-3)(x-2)\\P(x)=(x^{2}-3^{2})(x-2)\\P(x)=(x^{2}-9)(x-2)\\\\\boxed{\boxed{P(x)=x^{3}-2x^{2}-9x+18}}$

b)

$P(x)=(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3})(x-\alpha_{4})\\P(x)=(x-[3+i])(x-[3-i])(x-2)(x-[-1])\\P(x)=(x-3-i)(x-3+i)(x-2)(x+1)\\P(x)=([x-3]^{2}-i^{2})(x-2)(x+1)\\P(x)=(x^{2}-6x+9-[-1])(x^{2}+x-2x-2)\\P(x)=(x^{2}-6x+9+1)(x^{2}-x-2)\\P(x)=(x^{2}-6x+10)(x^{2}-x-2)\\P(x)=x^{4}-x^{3}-2x^{2}-6x^{3}+6x^{2}+12x+10x^{2}-10x-20\\P(x)=x^{4}-x^{3}-6x^{3}-2x^{2}+6x^{2}+10x^{2}+12x-10x-20\\\\\boxed{\boxed{P(x)=x^{4}-7x^{3}+14x^{2}+2x-20}}$

c)

$P(x)=(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3})(x-\alpha_{4})(x-\alpha_{5})\\P(x)=(x-[-3])(x-1)(x-0)(x-1)(x-3)\\P(x)=(x+3)(x-1)(x)(x-1)(x-3)\\P(x)=x(x+3)(x-3)(x-1)^{2}\\P(x)=x(x^{2}-3^{2})(x^{2}-2x+1)\\P(x)=x(x^{2}-9)(x^{2}-2x+1)\\P(x)=(x^{3}-9x)(x^{2}-2x+1)\\P(x)=x^{5}-2x^{4}+x^{3}-9x^{3}+18x^{2}-9x\\\\\boxed{\boxed{P(x)=x^{5}-2x^{4}-8x^{3}+18x^{2}-9x}}$