If α and β are zeros of the polynomial x2-ax+b then the value of α2α2β-β+β2β2α-αis1a2a2-4ba2-bb2ba2-4ba2-ba3aa2-4ba2-bb4b2a2-4bb2-aa - Maths - Polynomials

Question

I f   α   a n d   β   a r e   z
e r o s   o f   t h e   p o l y n o m i a l   x
2 - a x + b   t h e n   t h e   v a l u e   o f
  α 2 α 2 β - β + β 2 β 2 α
- α i s 1 a 2 a 2 - 4 b a 2 - b b 2 b a 2 - 4 b a 2 - b a 3 a
a 2 - 4 b a 2 - b b 4 b 2 a 2 - 4 b b 2 - a a

Varun Rawat · Accepted Answer

Let px = x2 - ax + bNow, α + β = -coefficient of xcoefficient of x2 = aαβ = constant termcoefficient of x2 = bNow, α2 + β2 = α + β2 - 2αβ = a2 - 2bNow, α - β2 = α2 + β2 - 2αβ = a2 - 2b - 2b = a2-4bNow, α2α2β - β + β2β2α - α2=α2α2 - β2β - β2α2 - β2α=α2 - β2α2β - β2α=α + βα - β×α3 - β3αβ=α + βα - βα - βα2 + β2 + αβαβ =α + βα - β2α2 + β2 + αβαβ =aa2 - 4ba2-2b+bb=aa2 - 4ba2-bb

I f α a n d β a r e z e r o s o f t h e p o l y n o m i a l x 2 - a x + b t h e n t h e v a l u e o f α 2 α 2 β - β + β 2 β 2 α - α i s 1 a 2 a 2 - 4 b a 2 - b b 2 b a 2 - 4 b a 2 - b a 3 a a 2 - 4 b a 2 - b b 4 b 2 a 2 - 4 b b 2 - a a