Question

Find a quadratic polynomial whose zeroes are:

$1,\sqrt{2}$

Answer 1

Definition:

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, $x=1$

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

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 $1,\sqrt{2}$

Therefore, the sum of the roots $(1+\sqrt{2})$
Product of the roots= $1\times \sqrt{2}=\sqrt{2}$
Calculation:

The required polynomial is

$x^2-(1+\sqrt{2})x+\sqrt{2}$

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 $k(x^2-(1+\sqrt{2})x+\sqrt{2})$,where $k$ is any non zero constant.

Final answer:

Hence the polynomial is $k(x^2-(1+\sqrt{2})x+\sqrt{2})$,where $k$ is any non zero constant