Question

If α and β are the zeros of the quadratic polynomial f (x) = x² + 3x -10, find the value of ( α³β + β³α ). 
1. 290
2. -318
3. 318
4. -290

Answer 1

Correct Answer - Option 4 : -290

Concept:

If α and β are the roots of equation , ax2 + bx + c =0 

Sum of roots (α + β) = \(\rm \frac{-b}{a}\)  

Product of roots  (αβ) = \(\rm \frac{c}{a}\)   

(x + y)² = x² + y² + 2xy .

Calculation:

Given: f (x) = x2 + 3x -10

Comparing f(x) with ax2 + bx + c =0 , we have , a = 1 , b= 3 and c= -10 . 

Now, sum of roots =  α + β = \(\rm \frac{-b}{a}\) = \(\rm \frac{-3}{1}\) = -3

And product of roots αβ = \(\rm \frac{c}{a}\) = \(\rm \frac{-10}{1}\) = -10 . 

Now, α3β + β3α = αβ ( α² + β² ) 

= αβ [ (α + β )² - 2αβ ] 

= (-10) [ (-3)² - ( 2 × (-10))]

= (-10) [ 9 + 20] 

= -290 .

The correct option is 4.