a factor (x-a)^k, k>1 yields a repeated zero x=a of multiplicity k
1. when k is odd, the graph crosses the x-axis at x=a
example:
f(x)=-4x^5+36x^3 ........... k=5 is odd
factor completely
f(x)=-4x^3(x^2-9)
f(x)=-4x^3(x^2-3^2)
f(x)=-4x^3(x-3)(x+3)
-4x^3(x-3)(x+3)=0
-4x^3=0 =>x=0........a repeated
(x-3)=0=>x=3
(x+3)=0=>x=-3
so, you have:
a repeated zero x=0 of multiplicity 3
zero x=3 of multiplicity 1
zero x=-3 of multiplicity 1
at each point (0,0),(-3,0), and (3,0) the graph crosses the x-axis
2.
when k is even, the graph touches the x-axis at x=a (but does not cross x-axis )
f(x)=(x-3)^4 .......... k=4 is even
find zeros:
(x-3)^4 =0
x-3=0
x=3..............of multiplicity 4
