Polynomial Functions and Complex Zeros
Alice White
9 min read
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Study Guide Overview
This study guide covers polynomial functions and complex zeros. It explores real, imaginary, and complex numbers, including the concept of conjugate pairs. The guide discusses linear factors, multiplicity, and the relationship between zeros and x-intercepts. Finally, it explains how to determine the degree of a polynomial using successive differences and how to identify even and odd functions.
#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)
Real numbers can be positive, negative, or zero.
#Imaginary Numbers
Imaginary numbers involve the imaginary unit i, where . They are written in the form bi, where b is a real number.
Examples:
- 2i
- -5i
- (1/3)i
Imaginary numbers are a subset of complex numbers where the real part is zero.
#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)
Complex numbers are the most general form of numbers we're dealing with here. They include both real numbers (when b = 0) and imaginary numbers (when a = 0).
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). 👌
Remember: Zeros are the x-values that make the polynomial equal to zero. They are also called roots.
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
Real zeros correspond to x-intercepts on the graph of the polynomial.
Multiplicity refers to how many times a ...
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