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Section 2.7 The Fundamental Theorem of Algebra

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Section 2.7 The Fundamental Theorem of Algebra

- What is the Fundamental Theorem of Algebra?
- Where do we use the Fundamental Theorem of Algebra?

The Fundamental

Theorem of Algebra

If f(x) is a polynomial of degree n

where n > 0, then

the equation f(x) = 0 has at least one root in the set of complex numbers.

If f(x)is a polynomial of degree nwhere

n >0, then the equation f(x) = 0has

exactly nsolutions provided that

each solution repeated twice

is counted as 2 solutions, etc…

solve each of the following polynomial equation,

determine how many solutions each equation has,

and classify each solution as rational, irrational or imaginary.

LETS

Ex. 1

2x− 1 = 0

Solution is: x = ½, one, rational

Ex. 2

x2 − 2 = 0

Solution:

- two, irrational

Ex. 3

x3 − 1 = 0

Solution: (x−1)(x2 + x + 1), x = 1 – one rational, and use Quadratic formula for

- two,

imaginary

So, how many solutions do following equations have? Find all zeros and classify them:

x3 + 3x2 + 16x + 48 = 0

(x + 3)(x2 + 16)= 0

x + 3 = 0, x2 + 16 = 0

x = −3, x2 = −16

x = − 3, x = ± 4i,

1 rational, 2 imaginary – 3 total

x4 + 5x2 - 36 = 0

(x2 - 4) (x2 + 9) = 0

x2 – 4 = 0, x2 + 9 = 0

x2 = 4, x2= − 9

x = ± 2, x = ± 3i,

2 rational, 2 imaginary – 4 total

Zeros are:

Next- write the complete factorization

of this polynomial function

Real zeros → can see on graph as x-intercepts

Imaginary zeros → cannot see on graph

ALSO, the graph is tangent to the x-axis at the repeated zero x = − 1, but crosses the x-axis at the zero x = 2.

Are there any pattern??

YES!

When a factor x – k of a function f is raised to an odd power,

the graph of f crosses the x-axis at x = k.

When a factor x – kof a function f is raised to an even power,

the graph of f is tangent (bounces off) to the x-axis at x = k.

Complex zeros always occur in conjugate pairs

“If a + bi is a zero, then a – bi is also”

This is The Complex Conjugates Theorem:

If f is a polynomial function with real coefficients,

and a + bi is an imaginary zero of f,

then a – bi is also a zero of f.

Irrational Conjugates Theorem:

If f is a polynomial function with rational coefficients,

and is a zero of f,

then is also a zero of f.

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros:

1.

2.

Are there any other ways to find the number

of positive and negative real zeros?

Let f(x) be a polynomial function with real coefficients.

- The number of positivereal zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number.
- The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.

This rule works even if some terms are missing!

PRACTICE!

- What is the Fundamental Theorem of Algebra?
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.

- What methods do you use to find the zeros of a polynomial function?
Possible rational zeros theorem (PRZ), the graph appearance, synthetic division, and quadratic formula.

- How do you use zeros to write a polynomial function?
If x = #, it becomes a factor (x ± #). Multiply factors together to find the equation.

Group work:

Open your book on pg. 142

and work in pairs to complete

## 16, 18, 26, 28, 35 – 49 odd

Section 2.8

Analysis and Graphing of Polynomial Functions

HOW to do it WITHOUT

a calculator???

- Find x-intercepts (zeros of the function)
- Find y-intercept.
- Determine end behavior (leading coefficient test)
- Plot points between and beyond the x-intercepts and y-intercept
- Draw the graph trough the plotted points so it has the appropriate end behavior.

- They are a local maximum or a local minimum
- A polynomial of degree nhas at mostn – 1 turning points
- If a polynomial of degree nhasndistinct real zeros, then the graph has exactlyn – 1turning points
- To find the turning points with the graphing calculator use maximum & minimum in CALC menu

Open your book on pg. 147 and work with your partner on ## 1 – 4 from “Guided practice” at the bottom of the page

Maximizing and minimizing

in problem solving….

Read Ex. 3 on pg. 147 and complete # 39 on pg. 149