Finding a Cubic Polynomial Using Zeros and Their Relationships
Video Explanation
Question
Find the cubic polynomial whose
- sum of its zeroes is \(3\),
- sum of the products of its zeroes taken two at a time is \(-1\),
- product of its zeroes is \(-3\).
Solution
Step 1: Write the Standard Form of a Cubic Polynomial
Let the zeroes of the required cubic polynomial be \[ \alpha,\; \beta,\; \gamma. \]
The general form of a monic cubic polynomial with these zeroes is:
\[ x^3 – (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \beta\gamma + \gamma\alpha)x – \alpha\beta\gamma \]
Step 2: Substitute the Given Values
Given:
\[ \alpha + \beta + \gamma = 3, \]
\[ \alpha\beta + \beta\gamma + \gamma\alpha = -1, \]
\[ \alpha\beta\gamma = -3. \]
Substituting these values in the standard form, we get:
\[ x^3 – 3x^2 – x + 3 \]
Answer
The required cubic polynomial is:
\[ \boxed{x^3 – 3x^2 – x + 3} \]