theakdad
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$$x^3 -5x^2+8x-4$$
What is the simplest way to factorize given equation? Thank you.
What is the simplest way to factorize given equation? Thank you.
wishmaster said:$$x^3 -5x^2+8x-4$$
What is the simplest way to factorize given equation? Thank you.
I do not understand the part "By sight"...chisigma said:By sight You notice that x = 1 and a solution of the equation...
$\displaystyle P(x) = x^{3} - 5\ x^{2} + 8\ x - 4 = 0\ (1)$
... so that (x-1) divides P(x)... then divide P(x) by (x-1) and obtain a second order polynomial that possibly can be further factorized...
Kind regards
$\chi$ $\sigma$
+I have to find the roots of the polynomial...so have to factorize it...Bacterius said:You could always try plotting it, it's not very rigorous mathematically but since polynomials are pretty well-behaved it's easy to spot roots most of the time.
wishmaster said:I have to find the roots of the polynomial...so have to factorize it...
wishmaster said:$$x^3 -5x^2+8x-4$$
What is the simplest way to factorize given equation? Thank you.
Yes,its obvious...MarkFL said:What chisigma was getting at, is that if you observe that given:
$$P(x)=x^3-5x^2+8x-4$$
and you see that:
$$P(1)=1-5+8-4=0$$
then you know that:
$$P(x)=(x-1)f(x)\implies f(x)=\frac{P(x)}{x-1}$$
where $f$ is of degree 2. Thus we can use division to determine $f$:
$$\begin{array}{c|rr}& 1 & -5 & +8 & -4 \\ 1 & & +1 & -4 & 4 \\ \hline & 1 & -4 & 4 & 0 \end{array}$$
Thus, we have found:
$$P(x)=(x-1)\left(x^2-4x+4\right)$$
And so now you only have a quadratic left to factor...can you continue?
wishmaster said:Yes,its obvious...
So i get $$(x-1) (x-2) (x-2)$$
i have three roots: X1 = 1 and X2,3 = 2
MarkFL said:Yes, that's correct...you have the root $x=1$ and the root $x=2$, which is a repeated root, or of multiplicity 2.
wishmaster said:$$x^3 -5x^2+8x-4$$
What is the simplest way to factorize given equation? Thank you.
wishmaster said:I was just wondering how do you guess the first root...
In my example there is a simple polynomial...but what to do when you have harder cases?