Download 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
x   and as x   , is governed
n
a
x
n
by
( 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 (  ) n – 1.
T.F.A.E. (The Following Are Equivalent)
If x =r is a root for a polynomial P(x) then:
1.
2.
3.
4.
P(r) = 0
(x-r) is a factor for P(x)
x=r is a zero for P(x), i.e. it is the value that makes P(x) =0
(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
a n  1.
Solution:
Since 1 – 3i is a root, then its conjugate must also be a root. This means 1+3i is also
a root.
SO P x   ( x  r1)( x  r 2)( x  r 3)
P( x)  ( x  2)( x  (1  3i )( x  (1  3i ))
P( x)  ( x  2)( x  1  3i )( x  1  3i )
P( x)  ( x  2)(( x  1)  3i )(( x  1)  3i )
Now remember that (a  bi )( a  bi )  a 2  b 2
so here for P( x) a  ( x  1), b  3
P( x)  ( x  2)(( x  1) 2  9)
P( x)  ( x  2)( x 2  2 x  1  9)
P( x)  ( x  2)( x 2  2 x  10)
P( x)  ( x 3  2 x 2  10 x  2 x 2  4 x  20)
P( x)  x 3  6 x  20