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Sagot :
EXPLANATION
Given the equation:
x^3 -2x^2 -5x + 10 = 0
Grouping terms:
[tex]=(x^3-2x^2)+(-5x+_{}10)[/tex]Factor out -5 from (-5x+10) and x^2 from (x^3-2x^2):
[tex]=x^2(x-2)-5(x-2)[/tex]Factor out common term x-2:
[tex]=(x-2)(x^2-5)[/tex]Factor x^2-5:
[tex]x^2-5=x^2-(\sqrt[]{5})^2[/tex]Apply difference of two squares formula:
[tex]x^2-y^2=(x+y)(x-y)[/tex][tex]x^2-(\sqrt[]{5})^2=(x+\sqrt[]{5})(x-\sqrt[]{5})[/tex]Grouping factor:
[tex]=(x-2)(x+\sqrt[]{5})(x-\sqrt[]{5})[/tex]Equaling to zero in order to get the roots:
[tex](x-2)(x+\sqrt[]{5})(x-\sqrt[]{5})=0[/tex]Using the zero factor principle:
[tex]\text{If ab=0, then a=0 or b=0}[/tex][tex]x-2=0\longrightarrow\text{ x=2 \lbrack{}First root\rbrack}[/tex][tex]x+\sqrt[]{5}=0\longrightarrow\text{ x=-}\sqrt[]{5}\text{ \lbrack{}Second root\rbrack}[/tex][tex]x-\sqrt[]{5}=0\longrightarrow\text{ x=}\sqrt[]{5}\lbrack Third\text{ Root\rbrack}[/tex]The roots are:
x=2, x=-sqrt(5) and x=sqrt(5)
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