Post on 10-Jul-2020
POLYNOMIAL FUNCTIONS Mr. Velazquez
Honors Precalculus
DEFINITION OF A POLYNOMIAL
Polynomials of degree 1 or higher have graphs that are smooth
and continuous.
• By smooth, we mean that the graphs contain only rounded curves
with no sharp corners, kinks or cusps
• By continuous, we mean that the graphs have no breaks or gaps,
and can be drawn without lifting your pencil
DEFINITION OF A POLYNOMIAL
END BEHAVIOR OF A POLYNOMIAL
END BEHAVIOR OF A POLYNOMIAL
ZEROS OF POLYNOMIALS
If 𝑓 is a polynomial function, then the values of 𝑥 for which 𝑓 𝑥 = 0 are called the zeros of 𝑓. These values of 𝑥 are also the x-intercepts of the graph of 𝑓, and the roots, or solutions, of the polynomial equation 𝑓 𝑥 = 0.
EXAMPLE: Find all zeros of 𝑓 𝑥 = 𝑥3 + 4𝑥2 − 3𝑥 − 12
By definition, these are the values of 𝑥 for which 𝑓 𝑥 = 0, so we simply set the function equal to zero, factor it, and set the factors equal to zero to solve for 𝑥:
Therefore, the zeros of 𝑓(𝑥) are:
EXAMPLES: Find all zeros for the following polynomial functions.
𝑓 𝑥 = 3𝑥2 + 16𝑥 − 12
𝑔 𝑥 = 𝑥3 + 2𝑥2 − 4𝑥 − 8
MULTIPLICITY OF ROOTS
MULTIPLICITY OF ROOTS
CLASSWORK, PART 1 (DO NOW!!)
On a separate sheet of paper (which should be kept for part 2 later), answer
the following question:
INTERMEDIATE VALUE THEOREM
Since 𝑓(𝑎) is negative and 𝑓 𝑏 is
positive, there must be some value of 𝑥between 𝑎 and 𝑏 for which 𝑓 𝑥 = 0
INTERMEDIATE VALUE THEOREM
Show that the function 𝑓 𝑥 = 𝑥3 − 2𝑥 − 5 has a real zero
between 𝑥 = 2 and 𝑥 = 3.
TURNING POINTS OF POLYNOMIALS
In general, a polynomial function of degree 𝑛 can have up to 𝑛 − 1 turning points.
Notice how the function on the left is a degree 5 polynomial and has 4 smooth turning points.
GRAPHING POLYNOMIAL FUNCTIONS
GRAPHING POLYNOMIAL FUNCTIONS
GRAPHING POLYNOMIAL FUNCTIONS
CLASSWORK, PART 2On the same sheet of paper as Part 1, sketch the
graph of the function 𝒇 𝒙 = 𝒙𝟑 − 𝟑𝒙𝟐 , and clearly label all zeros and y-intercepts.
HOMEWORK (for 2nd 9 weeks):
2.3 – Pg. 312, #2-46 (evens)