Transcript Notes 6.2 Polynomial & Linear Factors

5.2
Polynomial & Linear Factors
Learning goals
•graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and
using technology for more complicated cases
•graph polynomial functions, identifying zeros when
suitable factorizations are available, and showing end
behavior
Ex 1 Write in factored form, and then find the zeros.
3x  18 x  24 x
3
2
The following are equivalent statements about
a real number b and a polynomial P(x):
• (x – b) is a factor of P(x)
• b is a zero of the function
• b is a root (or solution) of P(x) = 0
• b is an x-intercept of the graph
So, for example…
Equivalent Statements about Polynomials
2
x  3x  4
•
– 4 is a solution of x  3 x  4  0
•
– 4 is an x-intercept of the graph
•
– 4 is a zero of
•
(x + 4) is a factor of x 2  3 x  4
2
y  x 2  3x  4
Ex 2 Find the zeros and graph
y   x  1 x 1 x  3
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
x
Ex 3 Find the zeros and graph
y    x  7  x  5 x  3
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
x
Ex 4 Write an equation in standard form with
zeros at 2, 3, and 0
Vocabulary
multiple zero : a repeated zero of a
polynomial that occurs because of a
repeated linear factor
multiplicity : the number of times a zero
occurs in a polynomial
Ex 5 For the following polynomial, find any
multiple zeros and state the multiplicity of
each f  x   x5  6 x4  9x3
Vocabulary
relative minimum and relative maximum :
peaks and valleys within the graph of a
polynomial that has several turning points
Ex 5 Find the relative minimum and maximum
f  x   3x  x  5x
3
2