Question

Find a quadratic polynomial whose zeroes are:

$2+\sqrt{3},2-\sqrt{3}$

Answer 1

Zeroes of a polynomial:

The zeros of a polynomial f(x) are all the values of x that make the polynomial equal to zero.

Example:

Let, $f\left(x\right)=3x^2-2x-1$

At, $x=1$,

$$\begin{gathered} f\left(1\right)=3\times 1^2-2\times 1-1 \\ orf\left(1\right)=3-2-1 \\ orf\left(1\right)=0 \end{gathered}$$

Then, $1$ is a zero of $f\left(x\right)$.

Formula:

We know that a quadratic polynomial whose zeroes are given can be represented as:

$x^2-(sumoftheroots)x+(productoftheroots)$

Explanation:

As the zeros are given $(2+\sqrt{3}),(2-\sqrt{3})$

Therefore, the sum of the roots=

$$\begin{gathered} (2+\sqrt{3})+(2-\sqrt{3}) \\ =2+\sqrt{3}+2-\sqrt{3} \\ =4 \end{gathered}$$

Product of the roots=

$$\begin{gathered} (2+\sqrt{3})(2-\sqrt{3}) \\ =(2^2-{\sqrt{3}}^2) \\ =4-3 \\ =1 \end{gathered}$$

Calculation:

The required polynomial is

$$\begin{gathered} x^2-\left(\right(2+\sqrt{3})+(2-\sqrt{3}\left)\right)x+\left(\right(2+\sqrt{3}\left)\right(2-\sqrt{3}\left)\right) \\ or{x}^2-4x+1 \end{gathered}$$

Conclusion:

Polynomial:

A polynomial is an expression of more than two algebraic terms, including the sum of several terms that contain different powers of the same variable(s).

Here, the polynomial is $k(x^2-4x+1)$,where $k$ is any non-zero constant.

Final answer:

Hence,the required polynomial is $k(x^2-4x+1)$,where $k$ is any non-zero constant.