Question

The polynomial equation of degree $5$ whose roots are the translates of the roots of
$x^5-2x^4+3x^3-4x^2+5x-6=0$ by $-2$ is

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

Answer 1

Answer: (B)

The polynomial equation of degree 5 whose roots are the translates of the roots of
$x^5-2x^4+3x^3-4x^2+5x-6=0$ by $-2$ is given
by $(x+2{)}^5-2(x+2{)}^4+3(x+2{)}^3$
$-4(x+2{)}^2+5(x+2)-6=0$
$\Rightarrow (x^5+10x^4+40x^3+80x^2+80x+32$
$-2\left(x^4+8x^3+24x^2+32x+16\right)$
$+3\left(x^3+6x^2+12x+8\right)-4$
$\left(x^2+4x+4\right)+5(x+2)-6=0$
$\Rightarrow x^5+8x^4+27x^3+46x^2+41x+12=0$