Question

If $a{x}^2+b{y}^2+c{z}^2+2ayz+2bzx+2cxy$ can be resolved into rational factors, then $a^3+b^3+c^3$ is

• A. $abc$
• B. $3abc$
• C. $-3abc$
• D. $2abc$

Answer 1

The correct option is B $3abc$

Let $f\left(x\right)=a{x}^2+b{y}^2+c{z}^2+2ayz+2bzx+2cxy$

$f\left(x\right)=z^2\{a\frac{x^2}{z^2}+b\frac{y^2}{z^2}+c+2a\frac{y}{z}+2b\frac{x}{z}+2c\frac{x}{z}\frac{y}{z}\}$

Let $X=\frac{x}{z},Y=\frac{y}{z}$

$f\left(x\right)=z^2\{a{X}^2+b{Y}^2+c{Z}^2+2aY+2bX+2cXY\}$ .....(i)

$f\left(x\right)$ can be resolved in rational factors if term in the bracket of (i) can be resolved in rational factors.

The condition for which is

$\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$\Rightarrow abc+2abc-a\times a^2-b\times b^2-c\times c^2=0\Rightarrow 3abc-a^3-b^3-c^3=0\Rightarrow a^3+b^3+c^3=3abc$

So, option B is correct.