If ${ D }_{ P }=\left| \begin{matrix} P & 15 & 8 \\ { P }^{ 2 } & 35 & 9 \\ { P }^{ 3 } & 25 & 10 \end{matrix} \right| ,$ then ${ D }_{ 1 }+{ D }_{ 2 }+{ D }_{ 3 }+{ D }_{ 4 }+{ D }_{ 5 }$ is equal to -

Given

$D_p=\begin{vmatrix} p & 15 & 8\\ p^2 & 35 & 9\\ p^3 & 25 & 10\end{vmatrix}$

$=p(350-225)-15(10p^2-9p^3)+8(25p^2-35p^3)$

$D_1=1(350-225)-15(10(1)^2-9(1)^3)+8(25(1)^2-35(1)^3)$

$=125-15.(1)+8(-10)$

$=125-80-15$

$=125-95$

$=30$

$D_2=2(350-225)-15(10(2)^2-9(2)^3)+8(25(2)^2-35(2)^3)$

$=2.(125)-15(40-72)+8(100-280)$

$=250+15(32)-1440$

$=730-1440$

$=-710$

$D_3=3(350-225)-15(10(3)^2-9(3)^3)+8(25(3)^2-35(3)^3)$

$=3.(125)-15(90-243)+8(225-945)$

$=375+15(153)-8(720)$

$=375+2,295-5760$

$=-3090$.

$D_4=4(350-225)-15(10(4)^2-9(4)^3)+8(25(4)^2-35(4)^3)$

$=4.(125)-15(160-576)+8(400-2240)$

$=500+6240-8.(1840)$

$=5740-14720$

$=-8980$

$D_5=5(350-225)-15(10(5)^2-9(5)^3)+8(25(5)^2-35(5)^3)$

$=5(125)-15(250-1125)+8(625-4375)$

$=625+13125+(-30,000)$

$=-16250$.

$\therefore D_1+D_2+D_3+D_4+D_5=30-710-3090-8980-16250$

$=-29000$.