We determine all zeros of the function y(x)=(x−1)(x−2)(x−3)+6.
y(x)=0⇔(x−1)(x−2)(x−3)+6=0⇔(x2−3x+2)(x−3)+6=0⇔x3−6x2+11x−6+6=0⇔x3−6x2+11x=0⇔x(x2−6x+11)=0⇔x(x2−6x+11)=0⇔x=0 or x2−6x+11=0
We determine the zeros of x2−6x+11: D=(−6)2−4⋅1⋅11=−8. Hence, this part has no zeros.
Consequently, the only zero of y(x) is x=0.
y(x)=0⇔(x−1)(x−2)(x−3)+6=0⇔(x2−3x+2)(x−3)+6=0⇔x3−6x2+11x−6+6=0⇔x3−6x2+11x=0⇔x(x2−6x+11)=0⇔x(x2−6x+11)=0⇔x=0 or x2−6x+11=0
We determine the zeros of x2−6x+11: D=(−6)2−4⋅1⋅11=−8. Hence, this part has no zeros.
Consequently, the only zero of y(x) is x=0.