Fundamental Theorem of Algebra Essentials
Master key concepts about polynomial roots, complex conjugates, and the Fundamental Theorem of Algebra.
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Questions Covered in This Set
8 cards to master
What does the Fundamental Theorem of Algebra guarantee?
Every non-constant polynomial of degree n has exactly n roots (counting multiplicity) in the complex numbers.
What is the Conjugate Root Theorem?
For polynomials with real coefficients, if a + bi is a root, then a - bi must also be a root.
If a degree-4 polynomial has roots 2 + 3i and x = 5, what else must be true?
The polynomial must also have 2 - 3i as a root (conjugate pair), plus one more root to total 4 roots.
What is multiplicity of a root?
The number of times a particular root appears; for example, (x - 2)³ means x = 2 has multiplicity 3.
How do you find roots of x² - 4x + 13 = 0?
Using the quadratic formula: x = (4 ± √(16 - 52))/2 = (4 ± 6i)/2 = 2 ± 3i
Why must complex roots come in conjugate pairs for real polynomials?
Due to properties of complex conjugation: substituting a complex number and its conjugate into a real polynomial both yield zero.
A polynomial has roots 1 + i, 1 - i, and 3 (multiplicity 2). What is its degree?
Degree 4 (two complex roots plus one root with multiplicity 2: 1 + 1 + 2 = 4)
How does a root with even multiplicity affect the graph?
The graph touches but does not cross the x-axis at that point; it 'bounces' off the axis.
