3.3 (1) Zeros of Polynomials

Multiplicities, zeros, and factors,

Oh my

PSAT ReviewLet’s review from sample test #4.

We’ll look at #31, 37, 38. Others?

PODFactor 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?

PODFactor into linear factors and find the zeros. Graph them to confirm

your zeros.

• 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)

1. 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 itWe’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 itWe’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 itFind 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 itFind 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 x-axis?

Use itFind 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 Multiplicity Degree of six 0 2 Six linear factors -2/3 1 Three zeros– all real 5/2 3 Crosses the x-axis 3 times

2. g(x) = (x2 + x - 12)3(x2 - 9) = (x + 4)3(x – 3)3(x – 3)(x + 3)

Zero Multiplicity Degree of eight -4 3 Eight linear factors 3 4 Three zeros– all real -3 1 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 f(x) = x - 1

g(x) = (x - 1)3

h(x) = (x - 1)5

Next, graph 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.