Chapter 2.2, Problem 64E

To determine

To find a polynomial of degree $n=2$ that has zeroes $x=-\sqrt{2},\sqrt{2}$ .

Answer to Problem 64E

  $x^2-2$

Explanation of Solution

Given:

Zeroes of polynomial $x=-\sqrt{2},\sqrt{2}$ and degree is $n=2$ .

Concept Used:

If $x=a$ is a zero of a polynomial, then $\left(x-a\right)$ is a factor of that polynomial.

Calculation:

To find a polynomial of degree $n=2$ that has zeroes $x=-\sqrt{2},\sqrt{2}$ . Use the concept If $x=a$ is a zero of a polynomial, then $\left(x-a\right)$ is a factor of that polynomial.

Thus, $\left(x-\sqrt{2}\right)$ and $\left(x+\sqrt{2}\right)$ are factors of the polynomial to be found.

Now, to find a polynomial that has degree 2, and has factors $\left(x-\sqrt{2}\right)$ and $\left(x+\sqrt{2}\right)$ multiply both these factors, as

  $\begin{array}{c}\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)=x^2-{\left(\sqrt{2}\right)}^2\\ =x^2-2\end{array}$

Thus, the polynomial of degree $n=2$ that has zeroes $x=-\sqrt{2},\sqrt{2}$ is $x^2-2$ .