A3 3.2b zeros of polynomials, multiplicity, turning points homework: p. 361 25-63 odd

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!

examples

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)

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.

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.

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

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