Section 5.4 The Fundamental Theorem of Algebra:
Let's put all of this together into a "big picture!"
The FTA (Fundamental Theorem of Algebra): If f is a polynomial of degree n, with n≥1, then f has at least one zero. That is, the equation f(x) =0 has at least one root, or solution. The zero of f may be a non­real complex number.
From this theorem some other important ones follow:
Linear Factors Theorem: For polynomial f(x), with degree n ≥1, f can be factored as f(x) = an(x­c1)(x­c2)...(x­cn) where c1, c2, ...cn are constants (possibly non­real constants and not necessarily distinct).
In other words, an nth degree polynomial can be factored as a product of n linear factors.
Turning Points: A turning point of a graph is where the behavior changes from increasing to decreasing or vice­versa. The graph of a polynomial f(x) with degree = n will have no more than n­1 turning points.
X­Intercepts: The graph of an nth degree polynomial has at most n x­intercepts.
Multiplicity of Zeros: (refer to previous notes for the rules...) Note that if the multiplicity k ≥2, the graph will "flatten out" near that intercept.
Example:
Conjugate Roots Theorem (Conjugate Pairs): If a + bi is a zero of f(x), then the conjugate pair a ­ bi is also a zero of f(x).
This means that f(x) has factors (x + (a + bi)) and (x ­ (a ­ bi)).
Examples:
Check out the Summaries of Polynomial Methods for finding zeros
and for graphing on page 414 in the textbook!
Mrs. Rogers' Expectations:
Finding Zeros of a Polynomial:
1) List all possible Rational Zeros (p/q list)
2) (sometimes) Use Descartes' Rule of Signs
3) Graph f(x) on your calculator to try to determine a zero to
test
4) Use long or synthetic division to verify the zero algebraically
and to reduce the polynomial by one degree
5) Repeat steps 3 and 4 until the polynomial is reduced to a
quadratic
6) Solve the quadratic to find the remaining zeros
7) List all zeros in a solution set using braces { }
Graphing a Polynomial Without a Calculator:
1) Use the Leading Term Test to determine the end behavior of
the graph as x⇒±∞
2) Find f(0) to determine the y-intercept
3) Use a given zero and division to reduce the polynomial and find
all zeros (x-intercepts)
4) Use the multiplicity of zeros rule to determine the graph
behavior at each zero (x-intercept)
5) Use n-1 at most turning points rule
6) Sketch a possible graph of the polynomial by using all of this
information