Transcript PCH (3.3)(1) Zeros of Poly 10

3.3 (1) Zeros of Polynomials
Multiplicities, zeros, and factors,
Oh my
PSAT Review
Let’s review from sample test #4.
We’ll look at #31, 37, 38. Others?
POD
Factor into linear factors and find the zeros. Graph them to confirm
your zeros.
1.
6x3 - 2x2 - 6x + 2
2.
5x3 - 30x2 - 65x
What do you notice about the number of linear factors and the
number of zeros?
POD
Factor into linear factors and find the zeros. Graph them to confirm
your zeros.
1.
6x3 - 2x2 - 6x + 2 = 2(3x3 - x2 - 3x + 1) = 2(x2(3x – 1) – (3x – 1)
= 2(x2 – 1)(3x-1) = 2(x + 1)(x – 1)(3x – 1)
2.
5x3 - 30x2 - 65x = 5x(x2 – 6x – 13)
= 5x(x – (3 + √22))(x – (3 – √22))
What do you notice about the number of linear factors and the
number of zeros?
The relationship between zeros
and factors
If we include real and complex zeros, and consider multiplicities of
zeros, there are the same number of zeros as there are linear
factors.
How does this relate to the degree of the polynomial?
What are other names for “zeros?”
x-intercepts are what type of zero?
Does this mean every linear factor represents an x-intercept?
What sorts of factors do we get if we limit them to real numbers?
Use it
We’ve seen the match up between linear factors and
zeros in the POD. Now, find zeros of f(x) and g(x)
with algebra and by graphing.
f(x) = x4 - 3x3 +2x2
g(x) = x5 - 4x4 +13x3
Use it
We’ve seen the match up
between linear factors
and zeros in the POD.
Now, find zeros of f(x)
and g(x) with algebra
and by graphing.
f(x) = x4 - 3x3 +2x2
How do they match up
here?
Use it
We’ve seen the match up
between linear factors and
zeros in the POD. Now,
find zeros of f(x) and g(x)
with algebra and by
graphing.
g(x) = x5 - 4x4 +13x3
How do they match up here?
Use it
Find f(x) with zeros at x = -5, 2, and 4. (How many of these could
we come up with? What would they look like? How many could
be third degree?)
Now, add to those zeros that f(3) = -24.
What does the equation become?
Use it
Find the zeros and their multiplicities of
1.
f(x) = x2(3x + 2)(2x - 5)3
2.
g(x) = (x2 + x - 12)3(x2 - 9)
What is the degree of each of these polynomials? How many linear
factors does each have? How many zeros does each have?
Are they all real? How many time does the graph cross the xaxis?
Use it
Find the zeros and their multiplicities of
1.
f(x) = x2(3x + 2)(2x - 5)3 = xx(3x + 2)(2x – 5)(2x – 5)(2x – 5)
Zero
0
-2/3
5/2
2.
Multiplicity
2
1
3
Degree of six
Six linear factors
Three zeros– all real
Crosses the x-axis 3 times
g(x) = (x2 + x - 12)3(x2 - 9) = (x + 4)3(x – 3)3(x – 3)(x + 3)
Zero
-4
3
-3
Multiplicity
3
4
1
Degree of eight
Eight linear factors
Three zeros– all real
Crosses the x-axis 3 times
Use it
Create your own. Write a polynomial function with an
odd number of real roots and a pair of imaginary
roots.
Give it with linear factors.
Give it with real number factors.
Graphs of multiplicities– review
On calculators, graph
Next, graph
f(x) = x - 1
g(x) = (x - 1)3
h(x) = (x - 1)5
f(x) = (x - 1)2
g(x) = (x - 1)4
h(x) = (x - 1)6
What do you notice about the exponents and the
graphs?
Graphs of multiplicities—review
f(x) = (x – 1)
g(x) = (x - 1)3
h(x) = (x - 1)5
f(x) = (x - 1)2
g(x) = (x - 1)4
h(x) = (x - 1)6
Graphs of multiplicities– review
In a graph of f(x) = (x - c)m, if c is a real number,
the graph will cross the x-axis at c if m is odd.
the graph will touch the x-axis at c, but not cross it,
if m is even.