Summary for Polynomial Functions
Summary for Polynomial Functions
FTA (Fundamental Theorem of Algebra)
A polynomial of degree n must have n roots.
Roots can be real numbers or imaginary numbers or a combination of both.
For real polynomials i.e., polynomials with Integer coefficients, imaginary roots always come in pairs.
For graphing polynomial functions, use the strategy outlined in class; (on my web page).
Features of Graphs of Polynomial Functions
The end behavior of the function, i.e., as [pic], is governed by [pic]( the term of the highest order).
If the multiplicity of the root is even then the graph touches the x-axis.
If the multiplicity of the root is odd, then the graph crosses the x-axis.
The number of turns in the graph of a polynomial function is at most ([pic] ) n – 1.
T.F.A.E. (The Following Are Equivalent)
If x =r is a root for a polynomial P(x) then:
1. P(r) = 0
2. (x-r) is a factor for P(x)
3. x=r is a zero for P(x), i.e. it is the value that makes P(x) =0
4. (r,0) is an x-intercept for the graph of P(x)
So for example : The answer is x= -2 , 1 – 3i. Find the polynomial of lowest power that has the above listed roots. Take [pic]
Solution:
Since 1 – 3i is a root, then its conjugate must also be a root. This means 1+3i is also a root.
[pic]
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