QUESTION

Remove the second term from the equation:${x^6} - 12{x^5} + 3{x^2} - 17x + 300 = 0$

Hint: To eliminate the second term we need the sum of the roots should be zero which can be provided by decreasing each root of the given equation by 2.

$f(x) = {x^6} - 12{x^5} + 3{x^2} - 17x + 300 = 0$
There are 6 roots of the equation and let the roots be ${r_1},{r_2},{r_3},{r_{4,}}{r_5}$ and ${r_6}$.
Considering general equation $f(x) = a{x^6} +b{x^5} + c{x^3} +d{x^2}+ex + f = 0$ sum of roots taken one at a time is $\dfrac{{ - b}}{a}$.
${r_1} + {r_2} + {r_3} + {r_4} + {r_5} + {r_6} = \dfrac{{ - ( - 12)}}{1} = 12$
$\Rightarrow {r_1} - 2 + {r_2} - 2 + {r_3} - 2 + {r_4} - 2 + {r_5} - 2 + {r_6} - 2 \\ \Rightarrow {r_1} + {r_2} + {r_3} + {r_4} + {r_5} + {r_6} - 12 \\ \Rightarrow 12 - 12 = 0 \\$
Hence, we need to decrease each of the roots of the $f(x) = 0$ by 2. So we need to replace x with ‘x+2’ in the equation. Transform the equation by putting $x = x + 2$, we get
${(x + 2)^6} - 12{(x + 2)^5} + 3{(x + 2)^2} - 17(x + 2) + 300 = 0$
$\Rightarrow {x^6} - 60{x^4} - 320{x^3} - 717{x^2} - 773x - 42 = 0$