I do not understand how we know a cubic polynomial such as
z^3 -3z^2 + 4z - 2 = 0
can be factored as
(z - 1)(z^2 - 2z +2) = 0
I can, of course, work it backwards but is this actually some kind of identity? and if so what is its derivation?
I do not understand how we know a cubic polynomial such as
z^3 -3z^2 + 4z - 2 = 0
can be factored as
(z - 1)(z^2 - 2z +2) = 0
I can, of course, work it backwards but is this actually some kind of identity? and if so what is its derivation?
At this sort of level, it's pretty much a case of trial and error.
Substitute values for z and hope that one of them gets you a zero value for the polynomial.
Start off by trying z = 0,1,2,-1,-2,3 etc. For this particular example you get a hit with z = 1 in which case z - 1 is a factor and you can now divide out to determine the quadratic.
Notice though that if for two of your trial values the polynomial changes sign then there has to be a zero between those two values. So for your example z = 0 and z = 2 gets you polynomial values of -2 and +2 in which case there has to be a value of z between 0 and 2 making the poynomial zero. (In general this need not be an integer value.)
If you get no joy from this method you might consider sketching a graph or graphs (usually graphs is better).
For this example rewrite the equation as $\displaystyle z^{3}-3z^{2}=2-4z$ and sketch the graphs of $\displaystyle y = z^{3}-3z^{2}$ and $\displaystyle y = 2 -4z.$ The values of z at which the graphs intersect will be the roots of the equation.
For this example this will tell you that there is a root somewhere between 1/2 and 3.
Hello, alyosha2!
I do not understand how we know a cubic polynomial such as: .$\displaystyle x^3 -3x^2 + 4x - 2 \:=\: 0$
can be factored as: .$\displaystyle (x - 1)(x^2 - 2z +2) \:=\:0$
We are given a polynomial equation: .$\displaystyle P(x) \:=\:0$
We are expected to know two theorems.
[1] If $\displaystyle P(a) = 0$, then $\displaystyle x = a$ is a root of the equation
. . .and $\displaystyle x-a$ is a factor of $\displaystyle P(x).$
[2] If $\displaystyle P(x)$ has a rational root, it is of the form: $\displaystyle x \,=\,\frac{n}{d}$
. . .where $\displaystyle n$ is a factor of the constant term
. . .and $\displaystyle d$ is a factor of the leading coefficient.
Our polynomial is: .$\displaystyle x^3 - 3x^2 + 4x - 2$
Its constant term is 2; its factors are: $\displaystyle \pm1,\:\pm2$
Its leading coefficient is 1; its factor are: $\displaystyle \pm1$
Hence, the only possible rational roots are: .$\displaystyle \pm1,\:\pm2$
So, we try each of them until one of them makes $\displaystyle P(x)$ equal to zero.
We find $\displaystyle x = 1$ is a zero of the polynomial.
. . Hence, $\displaystyle (x-1)$ is a factor.
Using long division, we have: .$\displaystyle x^3 - 3x^2 + 4x - 23 \:=\:(x-1)(x^2-2x+2)$