Question

If \(\alpha\), \(\beta\) are the zeroes of the quadratic polynomial \(px^2 + qx + r\), then find the value of \(\alpha^3\beta + \beta^3\alpha\).

• A. \(\frac{r(q^2 - 2pr)}{p^3}\)
• B. \(\frac{r(q^2 - 2pr)}{p^2}\)
• C. \(\frac{q(r^2 - 2pr)}{p^3}\)
• D. \(\frac{r(q^2 + 2pr)}{p^3}\)

Hint:

Whenever simplifying symmetric expressions of roots:
- Always factor out the greatest common factor first, which is \(\alpha\beta\) in this case.
- Keep the algebraic identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\) memorized, as it is one of the most frequently used relationships in polynomial questions.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a quadratic polynomial \(px^2 + qx + r\) with zeroes \(\alpha\) and \(\beta\). We need to determine the algebraic value of the expression \(\alpha^3\beta + \beta^3\alpha\) in terms of the coefficients \(p\), \(q\), and \(r\).

Step 2: Key Formula or Approach:
1. Use Vieta's formulas to establish relationships between the zeroes and coefficients:
\[ \alpha + \beta = -\frac{q}{p} \]
\[ \alpha\beta = \frac{r}{p} \]
2. Factor the target expression to express it entirely in terms of the sum (\(\alpha + \beta\)) and product (\(\alpha\beta\)) of the zeroes:
\[ \alpha^3\beta + \beta^3\alpha = \alpha\beta(\alpha^2 + \beta^2) \]
Substitute the algebraic identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\):
\[ \alpha^3\beta + \beta^3\alpha = \alpha\beta \left[ (\alpha + \beta)^2 - 2\alpha\beta \right] \]

Step 3: Detailed Explanation:
1. Write down Vieta's relations for the quadratic polynomial \(px^2 + qx + r\):
\[ \alpha + \beta = -\frac{q}{p} \]
\[ \alpha\beta = \frac{r}{p} \]
2. Express the target expression in terms of sum and product:
\[ \alpha^3\beta + \beta^3\alpha = \alpha\beta(\alpha^2 + \beta^2) \]
\[ \alpha^3\beta + \beta^3\alpha = \alpha\beta \left[ (\alpha + \beta)^2 - 2\alpha\beta \right] \]
3. Substitute the values of \(\alpha + \beta\) and \(\alpha\beta\) into the equation:
\[ \alpha^3\beta + \beta^3\alpha = \frac{r}{p} \left[ \left(-\frac{q}{p}\right)^2 - 2\left(\frac{r}{p}\right) \right] \]
4. Simplify the expression inside the bracket:
\[ \left(-\frac{q}{p}\right)^2 - 2\left(\frac{r}{p}\right) = \frac{q^2}{p^2} - \frac{2r}{p} \]
Take a common denominator of \(p^2\):
\[ \frac{q^2}{p^2} - \frac{2pr}{p^2} = \frac{q^2 - 2pr}{p^2} \]
5. Multiply this bracket by the outer factor \(\frac{r}{p}\):
\[ \alpha^3\beta + \beta^3\alpha = \frac{r}{p} \left[ \frac{q^2 - 2pr}{p^2} \right] = \frac{r(q^2 - 2pr)}{p^3} \]

Step 4: Final Answer:
The value of the expression is \(\frac{r(q^2 - 2pr)}{p^3}\), which corresponds to option (A).