Factoring and the Fundamental Theorem of Algebra
Master key concepts about polynomial factoring, zeros, and the Fundamental Theorem of Algebra.
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Questions Covered in This Set
10 cards to master
What is the fundamental connection between factors and zeros?
If (x - r) is a factor of a polynomial, then r is a zero (root) of that polynomial. Factors reveal zeros.
State the Fundamental Theorem of Algebra.
Every polynomial of degree n ≥ 1 with complex coefficients has exactly n complex roots (counting multiplicity).
What does the Rational Root Theorem tell us?
For polynomial p(x) with integer coefficients, any rational zero p/q must have p dividing the constant term and q dividing the leading coefficient.
How do complex roots appear in polynomials with real coefficients?
Complex roots always appear in conjugate pairs. If a + bi is a root, then a - bi must also be a root.
What is the sum of cubes factoring pattern?
a³ + b³ = (a + b)(a² - ab + b²)
What is the difference of cubes factoring pattern?
a³ - b³ = (a - b)(a² + ab + b²)
How does multiplicity affect the graph at a root?
Even multiplicity: the graph touches the x-axis. Odd multiplicity: the graph crosses the x-axis.
What are the steps in factoring by grouping?
Group terms strategically, factor out the GCF from each group, then factor out the common binomial factor.
If a cubic polynomial has roots 1, 2, and 3, what is its factored form (with leading coefficient a)?
p(x) = a(x - 1)(x - 2)(x - 3)
What should you do first when factoring any polynomial?
Factor out the greatest common factor (GCF) first.
