Question

If $x^3-2x^2-5x+6=0$ and $a,b,c$ are its roots, then match the following with their respective answers.

• A. $8$
• B. $14$
• C. $20$

Answer 1

Given equation is $x^3-2x^2-5x+6=0$ and
$a,b,c$ are the roots.
$\Rightarrow a+b+c=2,ab+bc+ca=-5,abc=-6$

$\left(i\right)(a+b+c{)}^2=a^2+b^2+c^2+2(ab+bc+ca)$
$\Rightarrow 2^2=a^2+b^2+c^2+2\times (-5)$
$\Rightarrow 4+10=a^2+b^2+c^2$
$\Rightarrow a^2+b^2+c^2=14$

$\left(ii\right)(a+b+c)(ab+bc+ca)=2\times (-5)$
$\Rightarrow a^{2}b+abc+a^{2}c+a{b}^2+b^{2}c+abc+abc+b{c}^2+c^{2}a=-10$
$\Rightarrow a^{2}b+a^{2}c+a{b}^2+b^{2}c+b{c}^2+c^{2}a+3abc=-10$
$\Rightarrow a^{2}b+a^{2}c+a{b}^2+b^{2}c+b{c}^2+c^{2}a+3(-6)=-10$
$\Rightarrow a^{2}b+a^{2}c+a{b}^2+b^{2}c+b{c}^2+c^{2}a=-10+18$
$\Rightarrow a^{2}b+a^{2}c+a{b}^2+b^{2}c+b{c}^2+c^{2}a=8$
$\Rightarrow \sum a^{2}b=8$

$\left(iii\right)(a+b+c)=2$
$a^2+b^2+c^2=14$
$\Rightarrow (a+b+c)(a^2+b^2+c^2)=2\times 14$
$\Rightarrow a^3+a{b}^2+a{c}^2+b{a}^2+b^3+b{c}^2+c{a}^2+c{b}^2+c^3=28$
$\Rightarrow a^3+b^3+c^3+a{b}^2+a{c}^2+b{a}^2+c{a}^2+c{b}^2=28$
$\Rightarrow a^3+b^3+c^3+8=28$
$\Rightarrow a^3+b^3+c^3=20$
$\Rightarrow \sum a^3=20$