Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively:

To find a quadratic polynomial given the sum and product of its zeroes, we can use the standard form of a quadratic polynomial. The general form of a quadratic polynomial is:

$p(x) = ax^2 + bx + c$

The relationships between the coefficients and the zeroes can be expressed as follows:

1. The sum of the zeroes $(\alpha + \beta)$ is given by $-\frac{b}{a}$.
2. The product of the zeroes $(\alpha \cdot \beta)$ is given by $\frac{c}{a}$.

Steps to find the quadratic polynomial

1. Define Sum and Product: Let the sum of the zeroes be $S$ and the product of the zeroes be $P$.

2. Use the relationships: From the above definitions, we have:

   • $S = -\frac{b}{a}$
   • $P = \frac{c}{a}$

3. Choose a value for $a$: Typically, we can take $a = 1$ for simplicity. This means:

   • $b = -S$
   • $c = P$

4. Construct the polynomial: After determining $b$ and $c$, substitute back into the polynomial form.

Example

Let's say the sum of the zeroes is $S = 5$ and the product of the zeroes is $P = 6$.

1. Given:

   • $S = 5$
   • $P = 6$

2. Using the relationships:

   • $b = -S = -5$
   • $c = P = 6$

3. The quadratic polynomial becomes: $p(x) = x^2 - 5x + 6$

Final Answer

The quadratic polynomial with the given sum and product of zeroes is:

$p(x) = x^2 - 5x + 6$