One of the roots of the equation 2000 x6+100 x5+10 x3+x-2=0 is of the form (m+√n/r), where m is non zero integer and n and r are relatively prime natural numbers. Find the value of m+n+r.

Question

One of the roots of the equation 2000 x6+100 x5+10 x3+x-2=0 is of the form (m+√n/r), where m is non zero integer and n and r are relatively prime natural numbers. Find the value of m+n+r. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

(2000 x6-2)+x(100 x4+10 x2+1)=0 2((10 x2)3-1)+x(100 x4+10 x2+1)=0 2(10 x2-1)(100 x4+100 x2+1)+x(100 x4+10 x2+1)=0 [2(10 x2-1)+x][100 x4+10 x2+1]=0 2 x2+x-2=0 D< 0 text no real solution.

Q. One of the roots of the equation $2000 x^{6} + 100 x^{5} + 10 x^{3} + x - 2 = 0$ is of the form $\frac{m + n}{r}$ , where $m$ is non zero integer and $n$ and $r$ are relatively prime natural numbers. Find the value of $m + n + r$ .

$(2000 x^{6} - 2) + x (100 x^{4} + 10 x^{2} + 1) = 0$
$2 ((10 x^{2})^{3} - 1) + x (100 x^{4} + 10 x^{2} + 1) = 0$
$2 (10 x^{2} - 1) (100 x^{4} + 100 x^{2} + 1) + x (100 x^{4} + 10 x^{2} + 1) = 0$
$[2 (10 x^{2} - 1) + x] [100 x^{4} + 10 x^{2} + 1] = 0$
$2 x^{2} + x - 2 = 0$
$D < 0 no real solution.$

Q. One of the roots of the equation 2000x6+100x5+10x3+x−2=0 is of the form rm+n​​, where m is non zero integer and n and r are relatively prime natural numbers. Find the value of m+n+r.

(2000x6−2)+x(100x4+10x2+1)=0 2((10x2)3−1)+x(100x4+10x2+1)=0 2(10x2−1)(100x4+100x2+1)+x(100x4+10x2+1)=0 [2(10x2−1)+x][100x4+10x2+1]=0 2x2+x−2=0 D<0 no real solution.

Q. One of the roots of the equation $2000 x^{6} + 100 x^{5} + 10 x^{3} + x - 2 = 0$ is of the form $\frac{m + n}{r}$ , where $m$ is non zero integer and $n$ and $r$ are relatively prime natural numbers. Find the value of $m + n + r$ .

$(2000 x^{6} - 2) + x (100 x^{4} + 10 x^{2} + 1) = 0$
$2 ((10 x^{2})^{3} - 1) + x (100 x^{4} + 10 x^{2} + 1) = 0$
$2 (10 x^{2} - 1) (100 x^{4} + 100 x^{2} + 1) + x (100 x^{4} + 10 x^{2} + 1) = 0$
$[2 (10 x^{2} - 1) + x] [100 x^{4} + 10 x^{2} + 1] = 0$
$2 x^{2} + x - 2 = 0$
$D < 0 no real solution.$