Polynomial questions (1 Viewer)

rand_althor · Oct 4, 2015

$$The polynomial $x^3-3x+2=0$ has roots $\alpha, \beta, \gamma$. Find $\alpha^4+\beta^4+\gamma^4$.$

What's the easiest way to do this?

Also, how do you do those questions where you are asked to find a polynomial with roots $$\alpha^2, \beta^2, \gamma^2$ , given that another polynomial (e.g. $x^3+2x^2-6x+1$ ) has roots $\alpha, \beta, \gamma$?$

Drsoccerball · Oct 4, 2015

rand_althor said:
$$The polynomial $x^3-3x+2=0$ has roots $\alpha, \beta, \gamma$. Find $\alpha^4+\beta^4+\gamma^4$.$

What's the easiest way to do this?

Also, how do you do those questions where you are asked to find a polynomial with roots $$\alpha^2, \beta^2, \gamma^2$ , given that another polynomial (e.g. $x^3+2x^2-6x+1$ ) has roots $\alpha, \beta, \gamma$?$

$Sub in roots into the equation$

$\alpha ^3 -3 \alpha +2=0$

$Multiply everything by the root$

$\alpha ^4 =3 \alpha ^2 -2 \alpha$

$\sum {\alpha ^4} = 3 \sum{\alpha ^2} -2 \sum{\alpha}$

[]

rand_althor · Oct 4, 2015

Drsoccerball said:
$Sub in roots into the equation$

$\alpha ^3 -3 \alpha +2=0$

$Multiply everything by the root$

$\alpha ^4 =3 \alpha ^2 -2 \alpha$

$\sum {\alpha ^4} = 3 \sum{\alpha ^2} -2 \sum{\alpha}$

[]

Ah okay. So then just do the same for beta and gamma, add those, rearrange, sub in values etc.?

rand_althor said:
Also, how do you do those questions where you are asked to find a polynomial with roots $$\alpha^2, \beta^2, \gamma^2$ , given that another polynomial (e.g. $x^3+2x^2-6x+1$ ) has roots $\alpha, \beta, \gamma$?$

Drsoccerball · Oct 4, 2015

rand_althor said:
Ah okay. So then just do the same for beta and gamma, add those, rearrange, sub in values etc.?

yes.
$x= \alpha ^2$

$\sqrt{x} = \alpha$

$$ Sub $ \alpha $ back into the original equation and now you have roots $ \alpha ^2 , \beta ^2 , \gamma ^2 .$

Paradoxica · Oct 4, 2015

Drsoccerball said:
yes.
$x= \alpha ^2$

$\sqrt{x} = \alpha$

$$ Sub $ \alpha $ back into the original equation and now you have roots $ \alpha ^2 , \beta ^2 , \gamma ^2 .$

$$A generalisation of the above$:$
$\noindent $If you are asked to find the polynomial with roots $ f(\alpha),$ then evaluate $ P(f^{-1}(x))$ and simplify.$$

rand_althor · Oct 4, 2015

Thanks! Do you know what the harder types of these questions are like? Like instead of $\alpha^2, \beta^2, \gamma^2$ , what else can there be?

Drsoccerball · Oct 4, 2015

rand_althor said:
Thanks! Do you know what the harder types of these questions are like? Like instead of $\alpha^2, \beta^2, \gamma^2$ , what else can there be?

In carrots trial it was $\frac{1}{\alpha ^2}$ Which was originally in the form $\frac{1}{( \alpha + \beta )^2}$

Polynomial questions (1 Viewer)

rand_althor

Active Member

Drsoccerball

Well-Known Member

rand_althor

Active Member

Drsoccerball

Well-Known Member

Paradoxica

-insert title here-

rand_althor

Active Member

Drsoccerball

Well-Known Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)