Search any question & find its solution
Question:
Answered & Verified by Expert
The polynomial equation of degree 5 whose roots are the translates of the roots of $x^5-2 x^4+3 x^3-4 x^2+5 x-6=0$ by -2 is
Options:
Solution:
1710 Upvotes
Verified Answer
The correct answer is:
$x^5+8 x^4+27 x^3+46 x^2+41 x+12=0$
The polynomial equation of degree 5 whose roots are the translates of the roots of $x^5-2 x^4+3 x^3-4 x^2+5 x-6=0$ by -2 is given $\begin{aligned} & \text { by }(x+2)^5-2(x+2)^4+3(x+2)^3 \\ &-4(x+2)^2+5(x+2)-6=0\end{aligned}$
$$
\begin{gathered}
\Rightarrow\left(x^5+10 x^4+40 x^3+80 x^2+80 x+32\right) \\
-2\left(x^4+8 x^3+24 x^2+32 x+16\right) \\
+3\left(x^3+6 x^2+12 x+8\right)-4 \\
\quad\left(x^2+4 x+4\right)+5(x+2)-6=0 \\
\Rightarrow \quad x^5+8 x^4+27 x^3+46 x^2+41 x+12=0
\end{gathered}
$$
$$
\begin{gathered}
\Rightarrow\left(x^5+10 x^4+40 x^3+80 x^2+80 x+32\right) \\
-2\left(x^4+8 x^3+24 x^2+32 x+16\right) \\
+3\left(x^3+6 x^2+12 x+8\right)-4 \\
\quad\left(x^2+4 x+4\right)+5(x+2)-6=0 \\
\Rightarrow \quad x^5+8 x^4+27 x^3+46 x^2+41 x+12=0
\end{gathered}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.