Question

If the equation $x^5-10a^3{x}^2+b^{4}x+c^5=0$ has 3 equals roots, then

• A. $2b^2-10a^3{b}^2+c^5=0$
• B. $6a^5+c^5=0$
• C. $2c^5-10a^3{b}^2+b^4{c}^5=0$
• D. $b^4=15a^4$

Answer 1

Answer: (B), (D)

The correct options are
B $6a^5+c^5=0$
D $b^4=15a^4$

$f\left(x\right)=x^5-10a^3{x}^2+b^{4}x+c^5$
$f^{{}^{\prime }}\left(x\right)=5x^4-20a^3-20a^{3}x+b^4$
$f^{\prime \prime }=20x^3-20a^3$
If $x=\alpha$ be a root that is repeated 3 times, then $f^{\prime \prime }\left(\alpha \right)=0$ and $f\left(\alpha \right)=0$
$\Rightarrow \alpha =a,5a^4-20a^4+b^4=0$
$a^5-10a^5+a{b}^4+c^5=0$
$\Rightarrow \alpha =a,b^4=15a^4$ and $6a^5+c^5=0$