**Polynomial function zeros:**

In math, a polynomial is an expression of finite length constructed from variables (also known as in determinates) and constants, using only the operations of addition,
subtraction, multiplication, and non-negative, whole-number exponents. For example, x^{2} − 4x + 7 is a polynomial, but x^{2} − 4/x + 7x^{3/2} is not, because its second
term involves division by the variable x and because its third term contains an exponent that is not a whole number.

Polynomial function zeros are a set of theorems aiming to find (or determine the nature) of the complex zeros of a polynomial function.

Note: i = square root (-1)

2 + i is a zero of p(y) = x^{4} - 2.x^{3} - 6.x^{2} + 22.x - 15

**Solution:**

**Step 1:** The zero 2 + i is an imaginary element

**Step 2:** p(y) has real coefficients.

**Step 3:** The conjugate 2 - i is also zero of p(y).

**Step 4**: The factored form of p(y)

p(y) = [x - (2 + i)] [x - (2 - i)]q(x)

**Step 5:** The expand the term of

[x - (2 + i)][x - (2 - i)] in p(y)

[x - (2 + i)][x - (2 - i)] = x^{2} -(2 + i)x -(2 - i)x + (2+i)(2-i)

= x^{2} - 4·x + 5

**Step 6:** q(x) dividing by p(y)

by x^{2} - 4·x + 5.

(x^{4} - 2·x^{3} - 6·x^{2} + 22·x - 15) / (x^{2} - 4·x + 5)

= x^{2} + 2·x - 3

**Step 7:** The factored form of p(y) is

p(x) = [x - (2 + i)][x - (2 - i)](x^{2} + 2·x - 3)

**Step 8:** The remaining 2 zeros of p(y)

**Step 9:** The solutions to the quadratic equation are

x^{2} + 2·x - 3 = 0

**Step 10:**To factor the quadratic equation of x^{2} + 2·x - 3 = 0 .

(x - 1)·(x + 3) = 0

**Step 11:** solutions

x = 1, x = -3

**Step 12:** The zeros of p(y)

2 + i , 2 - i, -3 and 1.

Problem 1: To calculate the function of p(x) = x^{2} + 5x +6

Solution: The zero of the function is x = -2 and -3

Problem 2: To calculate the function of P (z) = 2z + 4

Solution: The zero of the function is z= -2