If α, β are the zeroes of the quadratic polynomial px² + qx + r, then find the value of α³β + β³α.

α³β+β³α=(r(q²-2pr))/(p³)

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/2/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2026: question 21

Set in the CBSE Class 10 board exam of 2026, paper 30/2/1, section B, question 21, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21 · 2 marks · Very short answerPolynomials

If $α, β$ are the zeroes of the quadratic polynomial $p x^{2} + q x + r$ , then find the value of $α^{3} β + β^{3} α$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/2/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Take $α β$ common from both terms.

Hint 2

Write $α^{2} + β^{2}$ using $(α + β)^{2} - 2 α β$ , then use $α + β = - q / p$ and $α β = r / p$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

For $p x^{2} + q x + r$ , by the relation between zeroes and coefficients: $α + β = - \frac{q}{p}$ and $α β = \frac{r}{p}$
1 mark
$α^{3} β + β^{3} α = α β (α^{2} + β^{2}) = α β [(α + β)^{2} - 2 α β] = \frac{r}{p} [\frac{q ^{2}}{p ^{2}} - \frac{2 r}{p}]$
½ mark
$= \frac{r}{p} \cdot \frac{q ^{2} - 2 p r}{p ^{2}} = \frac{r ( q ^{2} - 2 p r )}{p ^{3}}$
½ mark

Answer: $α^{3} β + β^{3} α = \frac{r ( q ^{2} - 2 p r )}{p ^{3}}$

Where the marks are

Writes the sum and product of zeroes correctly (1 mark)
Factorises as $α β [(α + β)^{2} - 2 α β]$ and substitutes (½ mark)
Simplifies to $\frac{r ( q ^{2} - 2 p r )}{p ^{3}}$ (½ mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)
Topic: Relationship Between Zeros and Coefficients

Practise this chapterBoard-style questions from Polynomials, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 30/2/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2026: question 21

Hints

Step-by-step solution

Revise the chapter

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