The RootPowerSum( p, x, n ) command computes the sum of the $n$-th powers of the roots of the polynomial p in the indeterminate x.

Note that RootPowerSum( p, x, 0 ) is the same as the degree of p in x; RootPowerSum( p, x, 1 ) is the sum of the roots of p (as a polynomial in x); RootPowerSum( p, x, 2 ) is the sum of the squares of the roots of p; and so on.

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\:$

$p≔x^3-2x^2-5x+3$

$p≔x^3-2x^2-5x+3$

$\mathrm{RootPowerSum}\left(p\,x\,0\right)$

$3$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)$

$2$

$\mathrm{RootPowerSum}\left(p\,x\,2\right)$

$14$

$p≔\mathrm{expand}\left(\left(x-2\right){\left(x-1\right)}^2\right)$

$p≔x^3-4x^2+5x-2$

$\mathrm{RootPowerSum}\left(p\,x\,0\right)$

$3$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)$

$4$

$\mathrm{RootPowerSum}\left(p\,x\,2\right)$

$6$

$\mathrm{RootPowerSum}\left(p\,x\,3\right)$

$10$

$\mathrm{RootPowerSum}\left(p\,x\,4\right)$

$18$

$p≔x^{4}y-y^{2}x+xy-4$

$p≔x^{4}y-y^{2}x+xy-4$

$\mathrm{RootPowerSum}\left(p\,x\,0\right)$

$4$

$\mathrm{RootPowerSum}\left(p\,y\,0\right)$

$2$

$\mathrm{RootPowerSum}\left(p\,x\,3\right)$

$3y-3$

$\mathrm{RootPowerSum}\left(p\,y\,2\right)$

$\frac{x^7+2x^4+x-8}{x}$

$p≔\mathrm{expand}\left(\left(x-r\right)\left(x-s\right)\left(x-t\right)\right)$

$p≔-rst+rsx+rtx-r{x}^2+stx-s{x}^2-t{x}^2+x^3$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)$

$r+s+t$

$\mathrm{RootPowerSum}\left(p\,x\,2\right)$

$r^2+s^2+t^2$

$\mathrm{RootPowerSum}\left(p\,x\,30\right)$

$r^{30}+s^{30}+t^{30}$

$p≔a{x}^2+bx+c$

$p≔a{x}^2+bx+c$

$d≔\mathrm{discrim}\left(p\,x\right)$

$d≔-4ac+b^2$

$u≔\frac{-b+\mathrm{sqrt}\left(d\right)}{2a}$

$u≔\frac{-b+\sqrt{-4ac+b^2}}{2a}$

$v≔\frac{-b-\mathrm{sqrt}\left(d\right)}{2a}$

$v≔\frac{-b-\sqrt{-4ac+b^2}}{2a}$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)=\mathrm{normal}\left(u+v\right)$

$-\frac{b}{a}=-\frac{b}{a}$

$\mathrm{RootPowerSum}\left(p\,x\,2\right)=\mathrm{normal}\left(u^2+v^2\right)$

$-\frac{2ac-b^2}{a^2}=-\frac{2ac-b^2}{a^2}$

$\mathrm{RootPowerSum}\left(p\,x\,3\right)=\mathrm{normal}\left(u^3+v^3\right)$

$\frac{b\left(3ac-b^2\right)}{a^3}=\frac{b\left(3ac-b^2\right)}{a^3}$

$p≔\mathrm{expand}\left(\left(x-y\right)\left(x-2z\right)\left(x-yz\right)\right)$

$p≔-x^{2}yz+x{y}^{2}z+2xy{z}^2-2y^2{z}^2+x^3-x^{2}y-2x^{2}z+2xyz$

$\mathrm{RootPowerSum}\left(p\,x\,0\right)$

$3$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)$

$yz+y+2z$

$\mathrm{RootPowerSum}\left(p\,x\,2\right)$

$y^2{z}^2+y^2+4z^2$

$\mathrm{RootPowerSum}\left(\mathrm{mul}\left(x-r\Vert i\,i=1..12\right)\,x\,10\right)$

${\mathrm{r1}}^{10}+{\mathrm{r10}}^{10}+{\mathrm{r11}}^{10}+{\mathrm{r12}}^{10}+{\mathrm{r2}}^{10}+{\mathrm{r3}}^{10}+{\mathrm{r4}}^{10}+{\mathrm{r5}}^{10}+{\mathrm{r6}}^{10}+{\mathrm{r7}}^{10}+{\mathrm{r8}}^{10}+{\mathrm{r9}}^{10}$

$p≔\mathrm{expand}\left(\left(x-\mathrm{sqrt}\left(2\right)\right)\left(x-\mathrm{sqrt}\left(3\right)\right)\left(x-\mathrm{sqrt}\left(7\right)\right)\right)$

$p≔x^3-x^2\sqrt{7}-x^2\sqrt{3}+x\sqrt{3}\sqrt{7}-\sqrt{2}{x}^2+\sqrt{2}x\sqrt{7}+\sqrt{2}\sqrt{3}x-\sqrt{2}\sqrt{3}\sqrt{7}$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)$

$\sqrt{7}+\sqrt{3}+\sqrt{2}$

$\mathrm{RootPowerSum}\left(p\,x\,2\right)$

$12$

$\mathrm{RootPowerSum}\left(p\,x\,20\right)$

$282535322$

$p≔\mathrm{expand}\left(\left(x-\mathrm{sqrt}\left(2\right)\right)\left(x-\sin \left(1\right)\right)\left(x-\exp \left(t\right)\right)\right)$

$p≔x^3-x^2{\ⅇ}^t-x^2\sin \left(1\right)+x\sin \left(1\right){\ⅇ}^t-\sqrt{2}{x}^2+\sqrt{2}x{\ⅇ}^t+\sqrt{2}\sin \left(1\right)x-\sqrt{2}\sin \left(1\right){\ⅇ}^t$

$\mathrm{RootPowerSum}\left(p\,x\,1\right)$

${\ⅇ}^t+\sin \left(1\right)+\sqrt{2}$

$\mathrm{RootPowerSum}\left(p\,x\,4\right)$

${\left({\ⅇ}^t\right)}^4+{\sin \left(1\right)}^4+4$

The PolynomialTools[RootPowerSum] command was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.