A degree-4 polynomial with real coefficients has roots 2, -1, and 3 + 2i. What must the fourth root be?
A3 - 2i
B-3 + 2i
C-3 - 2i
DThere is no fourth root — a degree-4 polynomial can have fewer than 4 roots if one has high multiplicity
For a polynomial with real coefficients, complex roots always come in conjugate pairs: if a + bi is a root, then a - bi must also be a root. The conjugate of 3 + 2i is 3 - 2i, so that must be the fourth root. By the Fundamental Theorem, a degree-4 polynomial has exactly 4 roots counted with multiplicity — here all four are accounted for: 2, -1, 3 + 2i, and 3 - 2i. Option D is wrong: the theorem guarantees exactly n roots counted with multiplicity; a degree-4 polynomial always has 4.
Question 2 Multiple Choice
A student argues that the cubic polynomial p(x) = x³ + x² + x + 1 might have no real roots, since all three roots could be complex. Is this argument valid?
AYes — the roots of a cubic depend entirely on its coefficients, and complex roots are always possible
BNo — for a polynomial with real coefficients of odd degree, complex roots come in conjugate pairs, so at least one root must be real
CYes — x³ + x² + x + 1 has no positive real roots by Descartes' Rule, confirming they could all be complex
DNo — the Fundamental Theorem guarantees all roots of a real polynomial are real numbers
Complex (non-real) roots of real-coefficient polynomials come in conjugate pairs, each pair accounting for 2 roots. A degree-3 polynomial has 3 roots total. If any non-real complex roots exist, they come in pairs — but you cannot have 1.5 pairs. The remaining unpaired root must be real. Therefore, any odd-degree real polynomial has at least one real root. (In this specific case, x = -1 is a root.) Option D is wrong — real polynomials can certainly have non-real complex roots; they just can't all be non-real when the degree is odd.
Question 3 True / False
A degree-5 polynomial with real coefficients can have exactly 1 real root and 4 non-real complex roots.
TTrue
FFalse
Answer: True
Four non-real complex roots can occur as two conjugate pairs — (a + bi, a - bi) and (c + di, c - di) — which is perfectly consistent with real coefficients. Together with 1 real root, that accounts for all 5 roots that a degree-5 polynomial must have. This is a valid root structure. In contrast, a degree-5 polynomial cannot have exactly 2 real and 3 non-real complex roots, because 3 non-real roots cannot all be arranged in conjugate pairs.
Question 4 True / False
The polynomial x² - 6x + 9 = (x - 3)² has two distinct roots because it is a degree-2 polynomial and the Fundamental Theorem guarantees exactly two roots.
TTrue
FFalse
Answer: False
The Fundamental Theorem guarantees exactly 2 roots counted *with multiplicity*, not 2 distinct roots. The polynomial (x - 3)² has root x = 3 with multiplicity 2 — it is a double root, not two separate roots. There is only one distinct root value. Multiplicity counts how many times a factor (x - r) appears in the complete factorization. Confusing 'n roots counted with multiplicity' with 'n distinct roots' is a common error: a degree-n polynomial can have fewer than n distinct roots, but always exactly n when multiplicity is counted.
Question 5 Short Answer
A student argues: 'x² + 4 has no roots because there is no real number whose square equals -4.' How does the Fundamental Theorem of Algebra respond to this claim?
Think about your answer, then reveal below.
Model answer: The student is correct over the real numbers but wrong overall. The Fundamental Theorem guarantees that x² + 4 has exactly 2 roots in the complex numbers: x = 2i and x = -2i. These are non-real complex numbers (imaginary part ≠ 0) that satisfy the equation: (2i)² + 4 = -4 + 4 = 0. The theorem's key claim is that the complex number system is algebraically closed — every non-constant polynomial has exactly as many roots as its degree when complex numbers are allowed. Real numbers are not algebraically closed because polynomials like x² + 4 escape them.
The student's argument shows why real numbers alone are insufficient for polynomial algebra: some degree-n polynomials have fewer than n real roots (or none at all), breaking the elegant count that the Fundamental Theorem promises. Complex numbers were historically introduced precisely to fix this gap. Once you accept complex numbers, x² + 4 behaves like any other degree-2 polynomial — two roots, found by solving x² = -4, which gives x = ±√(-4) = ±2i. The theorem transforms a patchwork of special cases into a single universal guarantee.