Algebra 2 Quiz: Fundamental Theorem of Algebra Mastery Test

📚 Quick Study Guide

• 🔢 The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
• 📈 A polynomial of degree $n$ has exactly $n$ complex roots, counted with multiplicity.
• 💡 Complex roots of polynomials with real coefficients occur in conjugate pairs. If $a + bi$ is a root, then $a - bi$ is also a root.
• 📝 To find all roots of a polynomial, you can use synthetic division, factoring, and the quadratic formula.
• 🔍 Multiplicity refers to the number of times a root appears as a solution of the polynomial equation. For example, in $(x-2)^3$, the root 2 has a multiplicity of 3.

🧪 Practice Quiz

1. What does the Fundamental Theorem of Algebra guarantee?
1. A) Every polynomial has at least one real root.
2. B) Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
3. C) Every polynomial has exactly $n$ real roots.
4. D) Polynomials only have integer roots.
2. A polynomial has a degree of 5. How many complex roots does it have?
1. A) 2
2. B) 3
3. C) 4
4. D) 5
3. If $3 + 2i$ is a root of a polynomial with real coefficients, what must be another root?
1. A) $3 - 2i$
2. B) $-3 + 2i$
3. C) $-3 - 2i$
4. D) $2 - 3i$
4. What is the multiplicity of the root $x = 2$ in the polynomial $(x - 2)^4(x + 1)$?
1. A) 1
2. B) 2
3. C) 3
4. D) 4
5. Which of the following is NOT a direct consequence of the Fundamental Theorem of Algebra?
1. A) A polynomial of degree $n$ has $n$ roots, counting multiplicities.
2. B) Complex roots come in conjugate pairs for polynomials with real coefficients.
3. C) Every polynomial can be factored completely into linear factors over the complex numbers.
4. D) Polynomials always have rational roots.
6. A polynomial of degree 3 has roots $x = 1$ and $x = i$. What is the third root, assuming real coefficients?
1. A) $-i$
2. B) $-1$
3. C) $i$
4. D) $0$
7. How many complex roots does the polynomial $x^6 + 7x^4 + 12x^2$ have? (Hint: Factor out $x^2$ first)
1. A) 2
2. B) 4
3. C) 6
4. D) 8