Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.
Step-by-Step Solution
Key Concept: Use Vieta’s formulas for a quadratic polynomial $ax^2+bx+c=0$: sum of zeros $\alpha+\beta = -\frac{b}{a}$ and product of zeros $\alpha\beta = \frac{c}{a}$. Choose $a=1$ for simplicity and equate the given sum and product to find $b$ and $c$.
1. Let the required quadratic polynomial be \(ax^2+bx+c=0\).
2. For its zeros \(\alpha\) and \(\beta\), Vieta’s relations give:
$$\alpha+\beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}.$$
3. The problem states \(\alpha+\beta = -3\) and \(\alpha\beta = 2\).
4. Choose the simplest leading coefficient \(a=1\) (any non‑zero constant multiple would also work). Then the relations become:
$$-b = -3 \;\Rightarrow\; b = 3,$$
$$c = 2.$$
5. Substitute \(a=1,\; b=3,\; c=2\) into the general form:
$$x^2 + 3x + 2 = 0.$$
6. Hence the required quadratic polynomial (up to a non‑zero constant factor) is \(x^2 + 3x + 2\).
Correct Answer: x^2 + 3x + 2 = 0 (or any non‑zero constant multiple of this polynomial)