If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, find the value of a(α²/β + β²/α) + b(α/β + β/α)

Video Explanation

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Solution

Given polynomial:

f(x) = ax² + bx + c

Let α and β be the zeroes of the given quadratic polynomial.

Step 1: Write the Known Relations

For a quadratic polynomial ax² + bx + c:

α + β = −b/a

αβ = c/a

Step 2: Simplify Each Term

α²/β + β²/α

= (α³ + β³)/αβ

α³ + β³ = (α + β)³ − 3αβ(α + β)

∴ α²/β + β²/α

= {(α + β)³ − 3αβ(α + β)} / αβ

Similarly,

α/β + β/α = (α² + β²)/αβ

= {(α + β)² − 2αβ} / αβ

Step 3: Substitute the Values

Substitute α + β = −b/a and αβ = c/a

a(α²/β + β²/α)

= a[(α + β)³/αβ − 3(α + β)]

b(α/β + β/α)

= b[(α + β)²/αβ − 2]

Adding both expressions:

a(α²/β + β²/α) + b(α/β + β/α)

= a(α + β)³/αβ + b(α + β)²/αβ − 3a(α + β) − 2b

= [a(α + β)³ + b(α + β)²]/αβ − 3a(α + β) − 2b

Now,

a(α + β)³ + b(α + β)² = 0

So the expression becomes:

= −3a(α + β) − 2b

= −3a(−b/a) − 2b

= 3b − 2b

= b

Final Answer

The required value is b.

Conclusion

Thus, using the relationship between zeroes and coefficients of the quadratic polynomial, the value of a(α²/β + β²/α) + b(α/β + β/α) is b.