Question

Remove the second term from the equation:
$x^5+5x^4+3x^3+x^2+x-1=0$.

Answer 1

Given, $x^5+5x^4+3x^3+x^2+x-1=0$

There are 5 roots of the equation as it has a degree of 5. Let’s say $a,b,c,d$ and $e$

Sum of the roots = -5

To eliminate the $2^{nd}$ term we need sum of the roots should be $0$ which can be achieved by increasing each root of the given equation by 1.

$\therefore$ The we need to find the equation with roots $a+1,b+1,c+1,d+1$ and $e+1$

Transformed equation by inputting $x=x-1$ is $(x-1{)}^5+5(x-1{)}^4+3(x-1{)}^3+(x-1{)}^2+(x-1)-1=0$

$⟹x^5–7x^3+12x^2-7x=0$