Practice!

Sample Problem

Determine the multiplicity of each zero (write the number) and the number of turning points in the graph of f(t)=t5-7t3+10t.

(Order your zeros from least to greatest)

t= -√ ; multiplicity-

t= -√ ; multiplicity-

t= ; multiplicity-

t= √ ; multiplicity-

t= √ ; multiplicity-

number of turning points:

Solution

f(t)=t5-7t3+10t

f(t)=t(t4-7t2+10)

f(t)=t(t2-5)(t2-2)=t(t+√5)(t-√5)(t+√2)(t-√2)

zeroes: t= -√5, -√2, 0, √2, √5

t=-√5 has an odd multiplicity: multiplicity of 1 (t2-5)1

t=-√2 has an odd multiplicity: multiplicity of 1 (t2-2)1

t=0 has an odd multiplicity: multiplicity of 1 (t)1

t=√2 has an odd multiplicity: multiplicity of 1 (t2-2)1

t=√5 has an odd multiplicity: multiplicity of 1 (t2-5)1

number of turning points: 4 (A polynomial function of degree n has, at most, n-1 turning points.)