Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of...

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Holt McDougal Algebra 2 Fundamental Theorem of Algebra • How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots? • How do we identify all of the roots of a polynomial equation?

Transcript of Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of...

Page 1: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra

• How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots?

• How do we identify all of the roots of a polynomial equation?

Page 2: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra

You have learned several important properties about real roots of polynomial equations.

You can use this information to write polynomial function when given in zeros.

Page 3: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra Example 1: Writing Polynomial Functions

Write the simplest polynomial with roots –1, , and 4.2 3

1 xxP 23 x 4x

1 xxP 3 2x x14 8

33xxP 214x x8 23x x14 8 33xxP 211x x6 8

Page 4: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra Example 2: Writing Polynomial Functions

Write the simplest polynomial with roots –2, 2, and 4.

2x 2 xxP 4x

2 xxP 2x x6 8

3xxP 26x x8 22x x12 16 3xxP 24 x x4 16

Page 5: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra Example 3: Writing Polynomial Functions

Write the simplest polynomial with roots 0, , and 3.2 3

xxP 23 x 3x

xxP 3 2x x11 6

33xxP 211x x6

Page 6: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra Example 4: Writing Polynomial Functions

Write the simplest polynomial with roots –1, 5, and .3

3 5x 1 xxP 3x 3x

2xxP 2x 3x4 5 4xxP 23x 34x x12 25x 15 4xxP 34x 28x x12 15

Page 7: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra Example 5: Writing Polynomial Functions

Write the simplest polynomial with roots 2, 3i, and .5

,3i 5

2x 2 xxP 2x 529i 2 xxP 4x 24x 45

ix 3 2 xxP 5x 5x ix 3

5xxP x45 42 x 28x 9034x 5xxP 42 x 28x x45 9034x

Page 8: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra Example 6: Writing Polynomial Functions

Write the simplest polynomial with roots 1, 2 + i, and .3

,2 i 3

ix 2 1 xxP 3x 3x ix 2 2x 1 xxP 2x 3x2 ix x2 4 i2 ix i2 2i 1

1 xxP 2x x4 5 32 x 3xxP 2x x3 3 542 xx 5xxP 44x x15212x33x25x34x4x35x

15x1223x 5xxP 45x 36x 210x x27 15

Page 9: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra

Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

Page 10: Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Holt McDougal Algebra 2

Fundamental Theorem of Algebra

Lesson 4.2 Practice A


4.2. COMPLEX SOLUTIONS OF EQUATIONSskwon.org/2013fall/MLarTrig8_04_02.pdf · 2013. 6. 29. · The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra implies that a

4.2. COMPLEX SOLUTIONS OF EQUATIONSskwon.org/2013fall/MLarTrig8_04_02.pdf · 2013. 6. 29. · The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra implies that a

Topics in Analysis - University of Cambridgetwk/Topic.pdf · (which, contrary to its name, is a theorem of analysis). Theorem 2.8. [Fundamental Theorem of Algebra] If we work in the

Topics in Analysis - University of Cambridgetwk/Topic.pdf · (which, contrary to its name, is a theorem of analysis). Theorem 2.8. [Fundamental Theorem of Algebra] If we work in the

Holt McDougal Algebra 2 6-6 Fundamental Theorem of Algebra 6-6 Fundamental Theorem of Algebra Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson.

Holt McDougal Algebra 2 6-6 Fundamental Theorem of Algebra 6-6 Fundamental Theorem of Algebra Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson.

THE FUNDAMENTAL THEOREM OF ALGEBRA MADE EFFECTIVE: AN ELEMENTARY … roots... · 2012-09-20 · THE FUNDAMENTAL THEOREM OF ALGEBRA: A REAL-ALGEBRAIC PROOF 3 winding number (§4–§5),

THE FUNDAMENTAL THEOREM OF ALGEBRA MADE EFFECTIVE: AN ELEMENTARY … roots... · 2012-09-20 · THE FUNDAMENTAL THEOREM OF ALGEBRA: A REAL-ALGEBRAIC PROOF 3 winding number (§4–§5),

Holt Algebra 2 6-6 Fundamental Theorem of Algebra 6-6 Fundamental Theorem of Algebra Holt Algebra 2 Use the Fundamental Theorem of Algebra and its corollary.

Holt Algebra 2 6-6 Fundamental Theorem of Algebra 6-6 Fundamental Theorem of Algebra Holt Algebra 2 Use the Fundamental Theorem of Algebra and its corollary.

Holt McDougal Algebra 2 6-6 Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least.

Holt McDougal Algebra 2 6-6 Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least.

The Weak Fundamental Theorem of Algebra

The Weak Fundamental Theorem of Algebra

2.7 Apply the Fundamental Theorem of Algebra

2.7 Apply the Fundamental Theorem of Algebra

Section 2.4 The Fundamental Theorem of Algebra And Zeros of Polynomials.

Section 2.4 The Fundamental Theorem of Algebra And Zeros of Polynomials.

Ye Olde Fundamental Theorem of Algebra - MAA Sectionssections.maa.org/mddcva/MeetingFiles/Spring2012Meeting/TalkSlides/Parson.pdfThe Fundamental Theorem of Algebra A non-constant polynomial

Ye Olde Fundamental Theorem of Algebra - MAA Sectionssections.maa.org/mddcva/MeetingFiles/Spring2012Meeting/TalkSlides/Parson.pdfThe Fundamental Theorem of Algebra A non-constant polynomial

EULER’S FORMULA AND THE FUNDAMENTAL THEOREM OF ALGEBRA

EULER’S FORMULA AND THE FUNDAMENTAL THEOREM OF ALGEBRA

of the... · Web viewProofs of the Fundamental Theorem of Algebra. In his first proof of the Fundamental Theorem of Algebra, Gauss deliberately avoided using . imaginaries. When formulated

of the... · Web viewProofs of the Fundamental Theorem of Algebra. In his first proof of the Fundamental Theorem of Algebra, Gauss deliberately avoided using . imaginaries. When formulated

3.4 The Fundamental Theorem of Algebra - Franklin …cs.franklin.edu/~sieberth/MATH160/bookFiles/Chapter3/... ·  · 2006-12-273.4 The Fundamental Theorem of Algebra ... Examples

3.4 The Fundamental Theorem of Algebra - Franklin …cs.franklin.edu/~sieberth/MATH160/bookFiles/Chapter3/... ·  · 2006-12-273.4 The Fundamental Theorem of Algebra ... Examples

Section 2.7 The Fundamental Theorem of Algebra

Section 2.7 The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra and Finding Real Roots

Fundamental Theorem of Algebra and Finding Real Roots

COLLEGE ALGEBRA 3.2 Polynomial Functions of Higher Degree 3.3 Zeros of Polynomial Functions 3.4 Fundamental Theorem of Algebra.

COLLEGE ALGEBRA 3.2 Polynomial Functions of Higher Degree 3.3 Zeros of Polynomial Functions 3.4 Fundamental Theorem of Algebra.

FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence

FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence

2.7 Using the Fundamental Theorem of Algebra

2.7 Using the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra - Washington-Liberty...3.3 - Zeroes of Polynomials Fundamental Theorem of Algebra-Every polynomial of degree n will have n zeroes (real and complex/imaginary)

Fundamental Theorem of Algebra - Washington-Liberty...3.3 - Zeroes of Polynomials Fundamental Theorem of Algebra-Every polynomial of degree n will have n zeroes (real and complex/imaginary)

Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least.

Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least.