Question

If ∝, β are the roots of the equation $x^2$ - ax + b = 0, then ${\alpha }^4$+${\alpha }^3$β+${\alpha }^2$${\beta }^2$+α${\beta }^3$+${\beta }^4$

• A. a⁴ + 3a²b + b²
• B. a⁴ - 3a²b + b²
• C. a³ + 3a²b + b²
• D. None of these

Answer 1

The correct option is B

a⁴ - 3a²b + b²

α+β= a , αβ= b

therefore ${\alpha }^4$+${\alpha }^3$β+${\alpha }^2$${\beta }^2$+α${\beta }^3$+${\beta }^4$

=( α+β) (${\alpha }^3$+${\beta }^3$) + ${\alpha }^2$${\beta }^2$

=${(\alpha +\beta )}^2$[${(\alpha +\beta )}^2$ - αβ]+ ${\left(\alpha \beta \right)}^2$

= $a^2$($a^2$ – 3b) +$b^2$ =$a^4$ – 3$a^2$b+$b^2$