Question

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

• A. $x^{5}-8 x^{4}+27 x^{3}+46 x^{2}+41 x+12=0$
• B. $x^{5}+8 x^{4}+27 x^{3}+46 x^{2}+41 x+12=0$
• C. $x^{5}+6 x^{4}+28 x^{3}+46 x^{2}+41 x+12=0$
• D. $x^{5}+8 x^{4}+28 x^{3}+46 x^{2}+41 x+12=0$

Answer 1

Answer: (B)

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
by $(x+2)^{5}-2(x+2)^{4}+3(x+2)^{3}$
$-4(x+2)^{2}+5(x+2)-6=0$
$\Rightarrow \left(x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x+32\right.$
$-2\left(x^{4}+8 x^{3}+24 x^{2}+32 x+16\right)$
$+3\left(x^{3}+6 x^{2}+12 x+8\right)-4$
$\left(x^{2}+ 4 x+4\right)+5(x+2)-6=0$
$\Rightarrow x^{5}+8 x^{4}+27 x^{3}+46 x^{2}+41 x+12=0$