Complete the paragraph about roots of quadratic polynomials.
Check the final answer first, then review the worked steps.
Problem
Complete the paragraph about roots of quadratic polynomials.
Answer
Fundamental Theorem of Algebra
Step-by-step solution
- Identify the property: The problem states that any quadratic polynomial with real coefficients has complex roots, including repeated roots. This is a fundamental property in algebra.
- Recall relevant theorems: The behavior of polynomial roots is described by several theorems. For polynomials with real coefficients, a key theorem relates to the nature of their roots.
- Connect to 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. A corollary to this theorem, when applied to polynomials with real coefficients, is that complex roots must occur in conjugate pairs. This implies that a quadratic polynomial (degree 2) will always have two roots (counting multiplicity), and these roots can be real (which are also complex numbers with zero imaginary part) or a pair of complex conjugates. Therefore, all quadratic polynomials with real coefficients have complex roots.
- Complete the sentence: The statement "This follows from the..." refers to the theorem that guarantees the existence and nature of these roots. The most appropriate completion is the Fundamental Theorem of Algebra, which underpins these properties for polynomials.