Chapter 7: Functions
Imaginary Root Theorem: Imaginary roots of polynomial equations with real coefficients occur in conjugate pairs.
Quadratic Root Theorem: Quadratic roots of the form a + Ö b, where a and b are rational and Ö b is irrational, of polynomial equations with rational coefficients occur in conjugate pairs.
The Rational Root Theorem: If p and q are integers and p/q is a rational-number solution of a polynomial q equation anxn + a n-1 x n+1 + + a1x + ao = 0 with integer coefficients, then p is a factor of ao, and q is a factor of a.
When comes to graph the polynomial functions, the most commonly used way is to plug-in the numbers. But you have to remember following properties:
If f is a polynomial function such that f(a) > 0 and f(b) < 0, then there is a real number c between a and b such that f(c) = 0.
In the graph of a polynomial function f(x), a root corresponding to a point of tangency to the x-axis has an even multiplicity.
When comes to determine whether two functions are inverse, we can do it by graphing the two equations. But just remember the following property: Functions f and g are inverse functions if and only if f(g(x)) = g(f(x)) = x for all numbers x in the domains of both f and g.
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