Work Step by Step
See Section 3-2 (page 356) on Turning Points of Polynomial Functions. (If f is degree n, the graph has at most n-1 turning points) Also see The Leading Coefficient Test (p. 350, end behavior ) ( Odd $(\swarrow,\nearrow)$ or $(\nwarrow,\searrow)\quad$Even: $(\nwarrow,\nearrow)$ or $(\swarrow, \searrow)$ ) Also, see Multiplicity and x-lntercepts (odd: crosses the x-axis, even: touches and turns) ... graphs tend to flatten out near zeros with multiplicity greater than one. ------------------ This graph turns twice: n is at least 3. End behavior $(\nwarrow,\searrow)$: n is odd. Three zeros, all of odd multiplicity (crossing the x-axis). Note the flattening around x=0 ... multiplicity greater than 1 (at least 3) Each zero, with its multiplicity, generates linear factors of f. A polynomial of degree n can not have more than n linear factors, (The Linear Factorization Theorem), so, n is at least 1+3+1 =5.