Q2(iv):

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iv) $1, 1$

Given:

The sum of the zeroes of the quadratic polynomial ($\alpha + \beta$) = $1$.

The product of the zeroes of the quadratic polynomial ($\alpha \cdot \beta$) = $1$.

To Find:

A quadratic polynomial $p(x)$ of the form $ax^2 + bx + c$, where $a \neq 0$.

Step 1: Understanding the Relationship between Zeroes and Coefficients

For any quadratic polynomial $p(x) = ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$, the following relationships hold true:

Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$

Product of zeroes: $\alpha \cdot \beta = \frac{c}{a}$

Step 2: Formulating the General Quadratic Polynomial

A quadratic polynomial can be expressed in terms of the sum and product of its zeroes using the following identity:

$p(x) = k[x^2 - (\text{sum of zeroes})x + (\text{product of zeroes})]$

where $k$ is a non-zero real constant.

Step 3: Substituting the Given Values

Substitute the given values into the identity:

Sum of zeroes = $1$

Product of zeroes = $1$

$p(x) = k[x^2 - (1)x + (1)]$

$p(x) = k[x^2 - x + 1]$

Step 4: Determining the Polynomial

To obtain the simplest form of the polynomial, we assume $k = 1$ (as $k$ can be any non-zero real number, choosing $1$ provides the standard representation).

$p(x) = 1(x^2 - x + 1)$

$p(x) = x^2 - x + 1$

Verification (Optional):

For $p(x) = x^2 - x + 1$, here $a=1, b=-1, c=1$.

Sum of zeroes = $-\frac{b}{a} = -\frac{-1}{1} = 1$ (Matches given).

Product of zeroes = $\frac{c}{a} = \frac{1}{1} = 1$ (Matches given).

Final Answer: The required quadratic polynomial is $x^2 - x + 1$.