Objectives Fundamental Theorem of Algebra 6-6

Slides:

Advertisements

Similar presentations

Fundamental Theorem of Algebra

Advertisements

Notes 6.6 Fundamental Theorem of Algebra

Zeros of Polynomial Functions

Lesson 2.5 The Fundamental Theorem of Algebra. For f(x) where n > 0, there is at least one zero in the complex number system Complex → real and imaginary.

Finding Real Roots of Polynomial Equations 6-5

Lesson 4-1 Polynomial Functions.

9.9 The Fundamental Theorem of Algebra

6.6 The Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

Objectives Identify the multiplicity of roots.

Solving Higher Order Polynomials Unit 6 Students will solve a variety of equations and inequalities including higher order polynomials.

2.6 – Find Rational zeros Coach Bianco.

1 Using the Fundamental Theorem of Algebra.  Talk about #56 & #58 from homework!!!  56 = has -1 as an answer twice  58 = when you go to solve x 2 +

7.4 and 7.5 Solving and Zeros of Polynomials

Lesson 10.5 Factoring Objective: To factor a quadratic trinomial of the form Factoring a trinomial is the opposite of multiplying two binomials. Example:

3.6 Alg II Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify.

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots Identify all of the.

Holt Algebra Solving Quadratic Equations by Graphing and Factoring A trinomial (an expression with 3 terms) in standard form (ax 2 +bx + c) can be.

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

Section 3-6 Fundamental Theorem of Algebra

3.6 Complex Zereos. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra says that every polynomial with complex coefficients must have.

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.

Essential Questions How do we identify the multiplicity of roots?

By the end of this section, you will be able to: 1. Determine the number and type of roots for a polynomial equation; 2. Find the zeros of a polynomial.

7.5 Roots and Zeros Objectives: The student will be able to…

6-5 & 6-6 Finding All Roots of Polynomial Equations Warm Up: Factor each expression completely. 1. 2y 3 + 4y 2 – x 4 – 6x 2 – : Use factoring.

Chapter Fundamental Algebra theorem. Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least.

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.

5.6 The Fundamental Theorem of Algebra. If P(x) is a polynomial of degree n where n > 1, then P(x) = 0 has exactly n roots, including multiple and complex.

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

Solving Polynomials.

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.

1 What you will learn today…  How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function  How to use your.

Section 6: Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with.

Solve polynomial equations with complex solutions by using the Fundamental Theorem of Algebra. 5-6 THE FUNDAMENTAL THEOREM OF ALGEBRA.

Every polynomial P(x) of degree n>0 has at least one zero in the complex number system. N Zeros Theorem Every polynomial P(x) of degree n>0 can be expressed.

Factoring Polynomials.

Holt McDougal Algebra Fundamental Theorem of Algebra Intelligence is knowing that a tomato is a fruit; Wisdom is not putting it in a fruit salad.

Topic VII: Polynomial Functions Solving Polynomial Equations Roots and Zeros.

Algebra 2. Solve for x Algebra 2 (KEEP IN MIND THAT A COMPLEX NUMBER CAN BE REAL IF THE IMAGINARY PART OF THE COMPLEX ROOT IS ZERO!) Lesson 6-6 The Fundamental.

Rational Root Theorem and Fundamental Theorem of Algebra

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.

The Fundamental Theorem of Algebra

Rational Root Theorem and Fundamental Theorem of Algebra

Finding Real Roots of Polynomial Equations

Warm Up Identify all the real roots of each equation.

Warm Up Identify all the real roots of each equation.

Fundamental Theorem of Algebra

Lesson 7.2: Finding Complex Solutions of Polynomial Equations

7.5 Zeros of Polynomial Functions

Complex Numbers and Roots

Warm Up Identify all the real roots of each equation.

If a polynomial q(x) is divided by x – 4, the quotient is 2

Warm Up Identify all the real roots of each equation.

6-8 Roots and Zeros – Day 1 SWBAT

Apply the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

7.4 and 7.5 Solving and Zeros of Polynomials

3.4 Zeros of Polynomial Functions: Real, Rational, and Complex

Objectives Identify the multiplicity of roots.

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Warm Up Identify all the real roots of each equation.

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Fundamental Thm. Of Algebra

6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.

Fundamental Theorem of Algebra Roots/Zeros/X-Intercepts

Presentation transcript:

Objectives Fundamental Theorem of Algebra 6-6 Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the roots of a polynomial equation. Holt Algebra 2

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.

Example 1: Writing Polynomial Functions 2 3 Write the simplest polynomial with roots –1, , and 4. If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial. P(x) = x3 – x2 – 2x + 8 3 11

Check It Out! Example 1a Write the simplest polynomial function with the given zeros. –2, 2, 4 If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial. P(x) = x3– 4x2– 4x + 16

Write the simplest polynomial function with the given zeros. Check It Out! Example 1b Write the simplest polynomial function with the given zeros. 0, , 3 2 3 If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial. P(x) = x3– x2+ 2x 11 3

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.

Example 2: Finding All Roots of a Polynomial Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots. The polynomial is of degree 4, so there are exactly four roots for the equation. p = –36, and q = 1. Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots. Find the real roots at or near –1 and 4.

Example 2 Continued Test the possible real roots.

Example 2 Continued The polynomial factors into (x + 1)(x – 4)(x2 + 9) = 0. The solutions are 4, –1, 3i, –3i.

Check It Out! Example 2 Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots. Find the real roots at or near –5 and 1.

Check It Out! Example 2 Continued Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots. Find the real roots at or near –5 and 1.

Check It Out! Example 2 Continued The polynomial factors into (x + 5)(x – 1)(x2 + 4) = 0. The solutions are –5, 1, –2i, +2i).

HOMEWORK- Page 449 # 1-6, 11-19