Question & Answer
QUESTION

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

ANSWER Verified Verified
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.

Complete step-by-step answer:
$f(x) = {x^6} - 12{x^5} + 3{x^2} - 17x + 300 = 0$
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 $\dfrac{{ - b}}{a}$.
Comparing with given function we get b=-12 and a=1
\[{r_1} + {r_2} + {r_3} + {r_4} + {r_5} + {r_6} = \dfrac{{ - ( - 12)}}{1} = 12\]
 Hence to eliminate the second term, we need to reduce the sum to zero.
Now we can see if we subtract 2 from each roots, then the sum of roots will be zero i.e..,
\[
   \Rightarrow {r_1} - 2 + {r_2} - 2 + {r_3} - 2 + {r_4} - 2 + {r_5} - 2 + {r_6} - 2 \\
   \Rightarrow {r_1} + {r_2} + {r_3} + {r_4} + {r_5} + {r_6} - 12 \\
   \Rightarrow 12 - 12 = 0 \\
 \]
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$

Note: We can transform an equation into another by removing any number of terms. Since here we need to remove the second term we have to make the sum of the roots to zero. Make sure to find a correct value to add or subtract to the roots such that the sum of roots can be equal to zero.