Question

Find the quadratic polynomial whose sum and product of zeros are 6 and -2 respectively.

Hint:

The quadratic polynomial can be easily constructed from the sum and product of its zeros using the formula $p(x) = x^2 - (\text{sum of zeros})x + (\text{product of zeros})$.

Answer 1

Step 1: Recall the standard form of a quadratic polynomial.
A quadratic polynomial is given by the equation: \[ p(x) = ax^2 + bx + c \] The sum and product of the zeros $\alpha$ and $\beta$ of the polynomial are related to the coefficients as: - Sum of zeros: $\alpha + \beta = -\frac{b}{a}$ - Product of zeros: $\alpha \beta = \frac{c}{a}$
Step 2: Use the given sum and product of zeros.
We are given that the sum of zeros is 6 and the product is -2. Therefore: - Sum: $\alpha + \beta = 6 \quad \Rightarrow \quad -\frac{b}{a} = 6$ - Product: $\alpha \beta = -2 \quad \Rightarrow \quad \frac{c}{a} = -2$
Step 3: Write the quadratic polynomial.
Using the relationships above, the quadratic polynomial can be written as: \[ p(x) = x^2 - (\alpha + \beta)x + \alpha \beta = x^2 - 6x - 2 \]
Step 4: Conclusion.
Thus, the required quadratic polynomial is: \[ p(x) = x^2 - 6x - 2 \]