Exercise
Factor the polynomial equation
x^3-4x^2+5=0
Final Answer
Solution
x^3-4x^2+5=
=x^3-5x^2+x^2+5x-5x+5=
=x^3-5x^2+5x+x^2-5x+5=
=x(x^2-5x+5)+(x^2-5x+5)=
=(x^2-5x+5)(x+1)=
The first factor is a quadratic equation. its coefficients are:
a=1, b=-5, c=5
We solve it with the quadratic formula. Putting the coefficients in the formula gives us
x_{1,2}=\frac{5\pm \sqrt{{(-5)}^2-4\cdot 1\cdot 5}}{2\cdot 1}=
=\frac{5\pm \sqrt{5}}{2}
Hence, we get the solutions:
x_1=\frac{5+ \sqrt{5}}{2}
x_2=\frac{5- \sqrt{5}}{2}
Thus, the factorizing of the quadratic equation is
x^2-5x+5=
=(x-\frac{5+ \sqrt{5}}{2})(x-\frac{5- \sqrt{5}}{2})
All together we get
x^3-4x^2+5=
(x-\frac{5+ \sqrt{5}}{2})(x-\frac{5- \sqrt{5}}{2})(x+1)
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