Integrated Math 3: Improve Your Skills with Fundamental Theorem of Algebra Problems

Understanding the Fundamental Theorem of Algebra 🧠

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. In simpler terms, a polynomial of degree $n$ has exactly $n$ complex roots, counting multiplicity.

Key Concepts 🔑

• Complex Numbers: Numbers of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).
• Roots/Zeros: Values of $x$ for which the polynomial $P(x) = 0$.
• Multiplicity: The number of times a root appears as a solution of the polynomial equation.

Example Problem 1: Finding Roots 🔎

Consider the polynomial $P(x) = x^2 - 6x + 13$. Find all its roots.

1. Set the polynomial to zero: $x^2 - 6x + 13 = 0$
2. Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -6$, and $c = 13$.
3. Substitute the values: $x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(13)}}{2(1)}$ $x = \frac{6 \pm \sqrt{36 - 52}}{2}$ $x = \frac{6 \pm \sqrt{-16}}{2}$ $x = \frac{6 \pm 4i}{2}$ $x = 3 \pm 2i$

Therefore, the roots are $3 + 2i$ and $3 - 2i$.

Example Problem 2: Constructing a Polynomial 🏗️

Construct a polynomial with roots $2$, $-1$, and $1 + i$. Also, $1 - i$ must be a root since complex roots occur in conjugate pairs.

1. Write the factors: $(x - 2)$, $(x + 1)$, $(x - (1 + i))$, $(x - (1 - i))$
2. Multiply the factors: $P(x) = (x - 2)(x + 1)(x - 1 - i)(x - 1 + i)$
3. Simplify: $P(x) = (x^2 - x - 2)((x - 1)^2 - (i)^2)$ $P(x) = (x^2 - x - 2)(x^2 - 2x + 1 + 1)$ $P(x) = (x^2 - x - 2)(x^2 - 2x + 2)$ $P(x) = x^4 - 2x^3 + 2x^2 - x^3 + 2x^2 - 2x - 2x^2 + 4x - 4$ $P(x) = x^4 - 3x^3 + 2x^2 + 2x - 4$

Thus, the polynomial is $P(x) = x^4 - 3x^3 + 2x^2 + 2x - 4$.

Practice Problems ✍️

1. Find all roots of $P(x) = x^3 - x^2 + 4x - 4$.
2. Construct a polynomial with roots $3$, $-2i$, and $2i$.

Tips for Success ✅

• Master Complex Numbers: Understand operations with complex numbers.
• Practice Factoring: Improve your factoring skills.
• Use the Quadratic Formula: Be comfortable using the quadratic formula to find roots.

Code Example 💻

Here's a Python code snippet to find the roots of a polynomial using the numpy library:

import numpy as np

# Coefficients of the polynomial (highest degree first)
coefficients = [1, -6, 13]

# Find the roots
roots = np.roots(coefficients)

print(roots)