1. A3 3.2b zeros of polynomials, multiplicity, turning points
homework: p odd

2. zeros of a polynomial Zeros x-intercepts solutions
methods to find: factoring, graphing, long/synthetic division
remember the degree of the polynomial determines the number of solutions!

3. examples

4. multiplicity
if even multiplicity, then the graph “kisses” the x-axis and turns around. if the multiplicity is odd, then the graph crosses the x-axis.
revisit the previous example:
turn-around points: if the multiplicity is even, then the signs of the y-values do not change (no sign change) . if the multiplicity is odd, then the signs of the y-values change from positive to negative (or visa versa). max number of turning points is (n-1)

5. example
find the zeros of the polynomial. State each multiplicity. state whether the graph crosses or kisses the x-axis and turns around. support graphically.

6. intermediate value theorem
Let f be a polynomial with real coefficients. If f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c)=0.
example: show that the polynomial has a real zero between 2 and 3.

7. graphing higher degree polynomials by hand
determine end behavior using leading term test
find x-intercepts (y=0), solve by factoring
use even and odd multiplicities at specific zeros to sketch behavior of graph
find y-intercept (x=0)
use symmetry if applicable:
f(-x)=f(x) (even) symmetric about the y-axis
f(-x)=-f(x) (odd) point symmetry about the origin
6. use n-1 turning points to verify your graph

8. examples – graph by hand
whiteboard problem
graph by hand, showing all steps of the process…