x^3-6x^2+13x-10=0x3−6x2+13x−10=0
x^3-3(x)^2(2)+3(2)^2x+x-2^3-2=0x3−3(x)2(2)+3(2)2x+x−23−2=0
(x^3-3(x)^2(2)+3x(2)^2-2^3)+x-2=0(x3−3(x)2(2)+3x(2)2−23)+x−2=0
We can factorize using the polynomial identity that follows:
(a-b)^3= a^3-3a^2b+3ab^2+b^3(a−b)3=a3−3a2b+3ab2+b3
where in our case a=xa=x and b=2b=2
So,
(x-2)^3+(x-2)=0(x−2)3+(x−2)=0 taking x-2x−2 as common factor
(x-2)((x-2)^2+1)=0(x−2)((x−2)2+1)=0
(x-2)(x^2-4x+4+1)=0(x−2)(x2−4x+4+1)=0
(x-2)(x^2-4x+5)=0(x−2)(x2−4x+5)=0
x-2=0x−2=0 then x=2x=2
Or
x^2-4x+5=0x2−4x+5=0
delta=(-4)^2-4(1)(5)=16-20=-4<0δ=(−4)2−4(1)(5)=16−20=−4<0
delta <0rArrδ<0⇒ no root in R
