Find a polynomial of degree n that has only the given zero(s).

We will use the concept of factors of the polynomial to find the polynomial.

Answer: The standard polynomial will be (x - a\(_1\)) (x - a\(_2\)) (x - a\(_3\)).......(x - a\(_n\)) with a\(_1\),...a\(_n\) as their zeroes.

Let's see how we will use the concept of factors of the polynomial to find the polynomial.

Explanation:

When any polynomial is factorized, then the polynomial is segmented into its factors. When the factors are multiplied among themselves, then we get parent polynomial. 

Let's consider that a polynomial is given whose roots/zeros are a\(_1\), a\(_2\), a\(_3\),...... a\(_n\).

Then the polynomial will be equal to  (x - a\(_1\)) (x - a\(_2\)) (x - a\(_3\)).......(x - a\(_n\))  ------(1)

We can understand better by taking a random example.

Let us consider a polynomial of degree 3 with -4, 3, and -5 as their zeroes.

So, we can write x = - 4 , x = 3, x = -5

Therefore, the required polynomial will be (x + 4) (x - 3) (x + 5) [ From equation (1) ]

Hence, the polynomial will be equal to (x - a\(_1\)) (x - a\(_2\)) (x - a\(_3\)).......(x - a\(_n\)).