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Equations- Factorization of a polynomial equation – Exercise 5652

Exercise

Factor the polynomial equation

x^3-4x^2+5=0

Final Answer


(x-\frac{5+ \sqrt{5}}{2})(x-\frac{5- \sqrt{5}}{2})(x+1)=0

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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