Solved Examples: Fundamental Theorem of Algebra Explained (Algebra 2)

📚 Quick Study Guide

💡 Fundamental Theorem of Algebra: Every non-constant single-variable polynomial with complex coefficients has at least one complex root. In simpler terms, a polynomial of degree $n$ has $n$ complex roots, counting multiplicity. 🔢 Complex Root: A complex number ($a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit) that, when substituted for a variable in a polynomial equation, makes the equation true. 📈 Multiplicity: The number of times a root appears in the factorization of a polynomial. For example, in $(x-2)^3$, the root 2 has a multiplicity of 3. 🧭 Linear Factor Theorem: A polynomial $p(x)$ of degree $n$ can be factored into $n$ linear factors: $p(x) = a(x - c_1)(x - c_2)...(x - c_n)$, where $a$ is a constant and $c_1, c_2, ..., c_n$ are complex roots.

🧪 Practice Quiz

1. What does the Fundamental Theorem of Algebra state about the polynomial $p(x) = x^5 + 3x^2 - 7$?
   1. It has exactly 5 real roots.
   2. It has at least one complex root.
   3. It has no roots.
   4. It has exactly 3 complex roots and 2 real roots.
2. How many complex roots (counting multiplicity) does the polynomial $f(x) = (x - 3i)^2 (x + 1)$ have?
   1. 1
   2. 2
   3. 3
   4. 4
3. Which of the following is a possible root of the polynomial $g(x) = x^4 + 5x^2 + 4$?
   1. -1
   2. 0
   3. 1
   4. i
4. If a polynomial has a root of $2 - i$, what other root must it have if all coefficients are real?
   1. -2 - i
   2. 2 + i
   3. -2 + i
   4. -2
5. What is the degree of a polynomial that can be factored into $3(x - 2)(x + i)(x - 5i)(x+7)$?
   1. 3
   2. 4
   3. 5
   4. 6
6. A polynomial has roots at $x = 1$ (multiplicity 2) and $x = -3$. What is the smallest possible degree of the polynomial?
   1. 1
   2. 2
   3. 3
   4. 4
7. Which statement is NOT a consequence of the Fundamental Theorem of Algebra?
   1. Every polynomial of degree $n$ has $n$ roots in the complex number system.
   2. Every polynomial can be factored completely into linear factors over the complex numbers.
   3. Every polynomial has at least one real root.
   4. Every polynomial of odd degree has at least one real root.