Question

If $1,2,3$ and $4$ are the roots of the equation $x^4+a{x}^3+b{x}^2+cx+d=0$ then $a+2b+c$ is equal to

• A. $-25$
• B. $0$
• C. $10$
• D. $24$

Answer 1

The correct option is C

$10$

Explanation for correct answer:

Step-1: Express the given data and formula which is used.

Given, equation $x^4+a{x}^3+b{x}^2+cx+d=0$.

We know that, If $\alpha ,\beta ,\gamma \&\delta$ be the roots of equation. Then

$\Rightarrow$ $\alpha +\beta +\gamma +\delta =-\frac{b}{a},$

$\Rightarrow$ $\alpha ·\beta +\beta ·\gamma +\gamma ·\delta +\delta ·\alpha =\frac{c}{a}$

$\Rightarrow$$\alpha ·\beta ·\gamma +\alpha ·\beta ·\delta +\beta ·\gamma ·\delta +\alpha ·\gamma ·\delta =-\frac{d}{a}$

$\Rightarrow$ $\alpha ·\beta ·\gamma ·\delta =\frac{e}{a}$

Step-2: Finding required values.

$\Rightarrow$ $1+2+3+4=-a,$

$\Rightarrow$ $a=-10,$

$\Rightarrow$$1·2+2·3+3·4+1·3+1·4+2·4=b$

$\Rightarrow$ $b=35$

$\Rightarrow$ $1·2·3+1·2·4+2·3·4+1·3·4=-d$

$\Rightarrow$ $c=-50$

$\Rightarrow$ $1·2·3·4=d$

$\Rightarrow$ $d=24$

Therefore, the value of $a+2b+c=-10+2\times 35+\left(-50\right)=10$

Hence, correct answer is option $\left(C\right)$