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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.

Polynomial Function

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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