Polynomial Functions and Complex Zeros

#1.5 Polynomial Functions and Complex Zeros

Let's dive into the world of polynomial functions, exploring their zeros, multiplicities, and the fascinating interplay between real and complex numbers! 🐰

This section is crucial for understanding the behavior of polynomial functions, a key topic on the AP exam. Expect to see questions that combine concepts from this section with other areas.

#🔢 Real, Imaginary, and Complex Numbers

#Real Numbers

Real numbers are all the numbers you can find on a number line. They include:

• Integers (e.g., -3, 0, 5)
• Fractions (e.g., 1/2, -3/4)
• Decimals (e.g., 2.7, -0.5)
• Irrational numbers (e.g., √2, π)

#Imaginary Numbers

Imaginary numbers involve the imaginary unit i, where $i = \sqrt{-1}$. They are written in the form bi, where b is a real number.

Examples:

• 2i
• -5i
• (1/3)i

#Complex Numbers

Complex numbers combine real and imaginary parts, written in the form a + bi, where a and b are real numbers.

Examples:

• 3 + 4i
• -1 - 2i
• 5 + 0i (which is just the real number 5)

Image Courtesy of Brilliant

#Linear Factors and Multiplicities

A zero of a polynomial function p(x) is a value a such that p(a) = 0. If a is a real number, then (x - a) is a linear factor of p(x). 👌

If a polynomial has real coefficients, complex zeros always come in conjugate pairs. This means if a + bi is a zero, then a - bi is also a zero. 0️⃣

Multiplicity refers to how many times a ...