Question

Find a quadratic polynomial whose zeroes are:

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

Answer 1

Zeroes of a polynomial:

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

Example:

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

At, $x=1$,

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

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

Therefore, the product of the roots $\sqrt{2}\times \sqrt{3}=\sqrt{6}$

The sum of the roots $(\sqrt{2}+\sqrt{3})$

Calculation:

The required polynomial is $x^2-(\sqrt{2}+\sqrt{3})x+\sqrt{6}$

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 $k(x^2-(\sqrt{2}+\sqrt{3})x+\sqrt{6})$,where $k$ is non-zero constant.

Final Answer:

Hence the polynomial is $k(x^2-(\sqrt{2}+\sqrt{3})x+\sqrt{6})$