Lesson Objectives
• Learn how to write a polynomial function given certain conditions

## How to Use the Conjugate Zeros Theorem to Write a Polynomial Function

In this lesson, we will learn about the conjugate zeros theorem. The conjugate zeros theorem tells us that if we have a polynomial function with only real coefficients, then our non-real complex zeros come in conjugate pairs. This means if: a + bi is a zero, then a - bi is also a zero. We can use this fact to write a polynomial function. Let's look at an example.
Example #1: Write a polynomial function of least degree, having only real coefficients.
Zeros: -1, 1 + 4i
From the conjugate zeros theorem, we know that if (1 + 4i) is a zero, then (1 - 4i) is also a zero.
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$$f(x)=(x - (1 + 4i))(x + (1 - 4i))(x + 1)$$ $$f(x)=x^3 - x^2 + 15x + 17$$

#### Skills Check:

Example #1

Write a polynomial function of least degree, having only real coefficients.

Zeros: 1 - 7i, 3

A
$$f(x)=4x^3 + 5x + 1$$
B
$$f(x)=-x^3 + 5x^2 + 5x + 2$$
C
$$f(x)=x^3 + 9x^2 - 4x + 3$$
D
$$f(x)=2x^3 + x^2 + 7$$
E
$$f(x)=x^3 - 5x^2 + 56x - 150$$

Example #2

Write a polynomial function of least degree, having only real coefficients.

Zeros: 2 + i, 9

A
$$f(x)=7x^3 + 4x^2 + 3x - 1$$
B
$$f(x)=-x^3 - 5x^2 - 3x + 7$$
C
$$f(x)=2x^3 + 5x^2 + 4x - 1$$
D
$$f(x)=x^3 - 13x^2 + 41x - 45$$
E
$$f(x)=2x^3 - x^2 - 5$$