The third zero is -1

Introduction:
In this problem, we are given two zeroes of a polynomial p(x) and we need to find its third zero.

Given:
Zeroes of the polynomial p(x) are -1 and -3.

Solution:
Let's assume the third zero of the polynomial p(x) to be 'a'.

Using the sum of zeroes of a polynomial, we know that the sum of zeroes of a cubic polynomial is equal to the coefficient of x^2 divided by the coefficient of x^3. Hence, we have:

-1 - 3 + a = 5/1

Solving this equation, we get:

a = 9

Therefore, the third zero of the polynomial p(x) is 9.

Explanation:
A polynomial of degree n has exactly n zeroes. In this case, we are given two zeroes of the polynomial p(x), which means that p(x) has one more zero. Since p(x) is a cubic polynomial, it has exactly three zeroes.

To find the third zero, we use the sum of zeroes of a polynomial. This formula tells us that the sum of the zeroes of a polynomial of degree n is equal to the coefficient of x^(n-1) divided by the coefficient of x^n. For a cubic polynomial, this formula gives us:

sum of zeroes = -b/a

where 'a' is the coefficient of x^3 and 'b' is the coefficient of x^2.

In this case, the coefficient of x^2 is -5 and the coefficient of x^3 is 1. Hence, we have:

sum of zeroes = -(-5)/1 = 5

We know that the sum of the given zeroes (-1 and -3) is -1 - 3 = -4. Therefore, the sum of all three zeroes is:

sum of all zeroes = -4 + a = 5

Solving for 'a', we get:

a = 9

Therefore, the third zero of the polynomial p(x) is 9.