Write a polynomial function from its zeros

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 x₁, x₂, and x₃.

Then, the polynomial function can be found using f(x) = (x - x₁)(x - x₂)(x - x₃)

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)(x² - 4x - 4x + 16)

Simplify

f(x) = (x + 3)(x² - 8x + 16)

Use the distributive property

f(x) = x(x² - 8x + 16) + 3(x² - 8x + 16)

Use the distributive property

f(x) = x³ - 8x² + 16x + 3x² - 24x + 48

Combine like terms

f(x) = x³ - 8x² + 3x²+ 16x - 24x + 48

Simplify

f(x) = x³ - 5x² + -8x + 48

The function f(x) = x³ - 5x² + -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)(x² + x + 3x + 3)

Simplify

f(x) = (x - 2)(x² + 4x + 3)

Use the distributive property

f(x) = x(x² + 4x + 3) + -2(x² + 4x + 3)

Use the distributive property

f(x) = x³ + 4x² + 3x - 2x² - 8x - 6

Combine like terms

f(x) = x³ + 4x² - 2x²+ 3x - 8x - 6

Simplify

f(x) = x³ + 2x² - 5x - 6

The function f(x) = x³ + 2x² - 5x - 6 has zeros at 2, -3, and -1.

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