2.3--Polynomial Functions of Higher Degree with Modeling
Standard Form of a polynomial:
5x3 - 6x2 + 2x - 7
constant term
leading term
Terms are separated by addition and subtraction signs.
Coefficients are numbers that are multiplied by variables.
The degree of a polynomial is the value of its largest power.
Standard form requires the terms to be listed in
descending order, with the greatest exponent first.
The leading coefficient tells us which way the graph will
END UP going!!!
For example...
A)
y = + x3 + (blah blah blah...)
will either look
like this
B)
or this
y = - x3 + (blah blah blah...)
will either look
like this
or this
2.3--Polynomial Functions of Higher Degree with Modeling
The leading coefficient tells us which way the graph will
END UP going!!!
C)
y = + x4 + (blah blah blah...)
will either look
like this
D)
or this
y = - x4 + (blah blah blah...)
will either look
like this
or this
Find the zeros of each function. Work the problems
algebraically and check your answers graphically:
1)
f(x) = x3 - x2 - 12x
2)
f(x) = -2x3 + 5x2 + 2x - 5
2.3--Polynomial Functions of Higher Degree with Modeling
Multiplicity
3)
Sketch a graph of the function without using any
technology: (Then use Green Globs to see how we did!!!)
y = (x + 4)(x - 5)2
WINDOW:
y-min = -10
y-max = 10
4)
y = x4
WINDOW:
y-min = -10
y-max = 10
5)
y = x (x - 3)4
WINDOW:
y-min = -5
y-max = 25
6)
y = -(x + 2)3(x - 1)(x - 7)2
WINDOW:
y-min = -6000
y-max = 1000
2.3--Polynomial Functions of Higher Degree with Modeling
7)
Use the Intermediate Value Theorem to PROVE why every
polynomial function of odd degree has at least one real zero.
NOTE: Graphing & drawing pictures doesn't count as proof.
8) Bison Box Corporation has received an order to create cardboard
shipping containers which will hold 600 cm3 each. Squares are to be cut
from the corners of a 35 cm by 30 cm rectangular piece of cardboard,
and the flaps will be folded up to make an open-top box. What size
squares should be cut from the cardboard? (Round to 3 decimal places):
2.3--Polynomial Functions of Higher Degree with Modeling
Find a cubic function with the given zeros:
9)
2, 1, -5
10)
3, 2 + 5 , 2 - 5