Q) If α, ẞ are the zeroes of the quadratic polynomial p x 2 + qx + r then find the value of α 3 β + β 3 α. (Q 21 – 30/2/1 – CBSE 2026 Question Paper) Ans: Step 1: Let’s compare the given polynomial with standard quadratic polynomial, a x 2 + b x + c here, […]

If α, ẞ are the zeroes of the quadratic polynomial p x ^ 2 + qx + r

Q) If α, ẞ are the zeroes of the quadratic polynomial p x ² + qx + r then find the value of α ³ β + β ³α.

(Q 21 – 30/2/1 – CBSE 2026 Question Paper)

Ans:

Step 1: Let’s compare the given polynomial with standard quadratic polynomial, a x ² + b x + c

here, we get, a = p, b = q and c = r

Step 2: Next, we know that if α and β are zeroes of quadratic polynomial, then

Sum of the zeroes, α + β = $- \frac{b}{a} = - \frac {q}{p}$

and Product of zeroes, α . β = $\frac{c}{a} = \frac {r}{p}$

Step 3: We need to find value of α ³ β + β ³α

Let’s simplify it:

α ³ β + β ³α = α β (α ² + β ²)

∵ (a + b) ² = a ² + b ² + 2 a b

∴ a ² + b ² = (a + b) ² – 2 a b

∴ α ² + β ² = (α + β) ² – 2 α β

∴ α ³ β + β ³α = α β [(α + β) ² – 2 α β]

Step 4: Now we substitute the value of (α + β) and α . β from step 2

∴ α ³ β + β ³α = α β [(α + β) ² – 2 α β]

= $\frac {r}{p} ((- \frac {q}{p})^2 - 2 (\frac {r}{p}))$

= $\frac {r}{p} ((\frac {q^2}{p^2}) - \frac {2 r}{p})$

= $\frac {r}{p} (\frac {q^2 - 2 p r}{p^2})$

= $\frac {r(q^2 - 2 p r)}{p^3}$

Therefore, the value of α ³ β + β ³α for given polynomial is $\frac {r(q^2 - 2 p r)}{p^3}$ .

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