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The Fundamental Theorem of Algebra

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

German mathematician Carl Friedrich Gauss (1777-1855) first proved this theorem.

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.

Other wording……

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

Find all the zeros of the function by applying PRZs, Factoring, andQuadratic formula:

Zeros are:

Next- write the complete factorization

of this polynomial function

NOW, lets graph this function using your calculator!!!What do you observe???

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.

How about irrational zeros?

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.

How can we use the FTA to create the polynomial?

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

1.

2.

Descartes\' Rule of Signs

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!

Determine the number of possible positive real zeros, negative real zeros, and imaginary zeros for f(x)

PRACTICE!

Lets summarize what we have learned:

- 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.

Steps for Graphing of Polynomial Functions

- 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.

Special Points on the graph – turning points

- 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

Practice time!

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

When do we use turning points of a polynomial in real life??

Maximizing and minimizing

in problem solving….

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

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