Respuesta :
Given the roots of the equation to be:
[tex]x=-1,1,2\pm\sqrt[]{5}[/tex]Therefore, the factors are:
[tex](x+1),(x-1),(x-2+\sqrt[]{5}),(x-2-\sqrt[]{5})[/tex]To get the polynomial, we will multiply the factors together:
[tex]\Rightarrow(x+1)(x-1)(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})[/tex]Expanding in pairs, we can have:
[tex]\Rightarrow\lbrack(x+1)(x-1)\rbrack(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})[/tex]Recall the Difference of Two Squares given to be:
[tex]a^2-b^2=(a-b)(a+b)[/tex]Hence, the first pair becomes:
[tex](x+1)(x-1)=x^2-1^2=x^2-1[/tex]Hence, the polynomial becomes:
[tex]\Rightarrow(x^2-1)\lbrack(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})\rbrack[/tex]Expanding the pair:
[tex](x-2+\sqrt[]{5})(x-2-\sqrt[]{5})=x^2-2x-x\sqrt[]{5}-2x+4+2\sqrt[]{5}+x\sqrt[]{5}-2\sqrt[]{5}-5=x^2-4x-1[/tex]Hence, the expression becomes:
[tex]\Rightarrow(x^2-1)(x^2-4x-1)[/tex]Expanding, we get:
[tex]\Rightarrow x^4-4x^3-x^2-x^2+4x+1=x^4-4x^3-2x^2+4x+1^{}[/tex]Therefore, the polynomial is:
[tex]\Rightarrow x^4-4x^3-2x^2+4x+1[/tex]