Question

If one factor of $a^4+a^2{b}^2+b^4$ is $a^2+b^2+ab$, then the other factor is

• A. $a^3+b^3+c^3$
• B. $a^2+b^2-ab$
• C. $a^2+b^2+c^2$
• D. $a^2+b^2+ab$

Answer 1

Answer: (B)

$a^2+b^2-ab$

$a^4+a^2{b}^2+b^4=a^4+a^2{b}^2+b^4+a^2{b}^2-a^2{b}^2$

$=a^4+b^4+2a^2{b}^2-a^2{b}^2$

$=(a^2+b^2{)}^2-a^2{b}^2$............... using $(a+b{)}^2$ formula

$=(a^2+b^2{)}^2-(ab{)}^2$

$=\left(\right(a^2+b^2)+ab)\left(\right(a^2+b^2)-ab)$ .........using $(a^2-b^2)$ formula

$\therefore$ The factors of $a^4+a^2{b}^2+b^4$ are $(a^2+b^2+ab)$ and $(a^2+b^2-ab)$.