Learn to write a polynomial function from its zeros with a couple a good examples.

The goal is to write a linear factor for each zero and then multiply the linear factors. Suppose the zeros of a function are x_{1}, x_{2}, and x_{3}.

Then, the polynomial function can be found using f(x) = (x - x_{1})(x - x_{2})(x - x_{3})

**Example #1:**

Write a polynomial function in standard form with zeros at -3, 4, and 4.

f(x) = (x + 3)(x - 4)(x - 4)

Multiply (x - 4) and (x - 4)

f(x) = (x + 3)(x^{2} - 4x - 4x + 16)

Simplify

f(x) = (x + 3)(x^{2} - 8x + 16)

Use the distributive property

f(x) = x(x^{2} - 8x + 16) + 3(x^{2} - 8x + 16)

Use the distributive property

f(x) = x^{3} - 8x^{2} + 16x + 3x^{2} - 24x + 48

Combine like terms

f(x) = x^{3} - 8x^{2} + 3x^{2}+ 16x - 24x + 48

Simplify

f(x) = x^{3} - 5x^{2} + -8x + 48

The function f(x) = x^{3} - 5x^{2} + -8x + 48 has zeros at -3, 4, and 4.

**Example #2:**

Write a polynomial function in standard form with zeros at 2, -3, and -1.

f(x) = (x - 2)(x + 3)(x + 1)

Multiply (x + 3) and (x + 1)

f(x) = (x - 2)(x^{2} + x + 3x + 3)

Simplify

f(x) = (x - 2)(x^{2} + 4x + 3)

Use the distributive property

f(x) = x(x^{2} + 4x + 3) + -2(x^{2} + 4x + 3)

Use the distributive property

f(x) = x^{3} + 4x^{2} + 3x - 2x^{2} - 8x - 6

Combine like terms

f(x) = x^{3} + 4x^{2} - 2x^{2}+ 3x - 8x - 6

Simplify

f(x) = x^{3} + 2x^{2} - 5x - 6

The function f(x) = x^{3} + 2x^{2} - 5x - 6 has zeros at 2, -3, and -1.