Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this.

Published byBlanche Hawkins Modified over 8 years ago

Similar presentations

Presentation on theme: "Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this."— Presentation transcript:

1 Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this lesson, you will learn how to find the roots of a polynomial equation with integer coefficients that is not factorable. 1 2.3.4: The Rational Root Theorem

2 Key Concepts You can find the possible rational roots of a polynomial equation with integer coefficients using the Rational Root Theorem. This theorem states that if a polynomial has rational roots, then those roots will be the ratio of the factors of the constant term to the factors of the leading coefficient written in lowest terms. Be sure to include all factors of both the constant term and the leading coefficient, including both the positive and negative factors. It is not necessary to repeat duplicate ratios. 2 2.3.4: The Rational Root Theorem

3 Key Concepts, continued 3 2.3.4: The Rational Root Theorem Rational Root Theorem If the polynomial p(x) has integer coefficients, then every rational root of the polynomial equation p(x) = 0 can be written in the form, where p is a factor of the constant term of p(x) and q is a factor of the leading coefficient of p(x).

4 Key Concepts, continued It is possible for polynomial equations to have no rational roots; polynomial equations could also have irrational roots as stated in the Irrational Root Theorem. 4 2.3.4: The Rational Root Theorem Irrational Root Theorem If a polynomial p(x) has rational coefficients and is a root of the polynomial equation p(x) = 0, where a and b are rational and is irrational, then is also a root of p(x) = 0.

5 Key Concepts, continued Recall that it is also possible for an equation to have imaginary roots in the form a + bi and a – bi, referred to as complex conjugates. If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a – bi is also a root. 5 2.3.4: The Rational Root Theorem

6 Common Errors/Misconceptions misinterpreting the Rational Root Theorem as stating that all zeros of a polynomial function must be rational not identifying both the positive and negative factors of the constant term and leading coefficients not finding all factors of a polynomial function 6 2.3.4: The Rational Root Theorem

7 Guided Practice Example 2 Find all of the roots to the polynomial equation 4x 3 – x 2 + 36x – 9 = 0. 7 2.3.4: The Rational Root Theorem

8 Guided Practice: Example 2, continued 1.List all the possible rational roots. The Rational Root Theorem states that the only possible rational roots of the equation must be ratios of the factors of the constant term and the factors of the leading coefficient. The constant term is –9 and the leading coefficient is 4. The factors of the constant term, –9, are ±1, ±3, and ±9. The factors of the leading coefficient, 4, are ±1, ±2, and ±4. 8 2.3.4: The Rational Root Theorem

9 Guided Practice: Example 2, continued Use these factors to set up the ratios. 9 2.3.4: The Rational Root Theorem Ratio of the factors of the constant term and the factors of the leading coefficient Write a separate ratio for each of the factors.

10 Guided Practice: Example 2, continued Simplify each ratio. 10 2.3.4: The Rational Root Theorem

11 Guided Practice: Example 2, continued 2.Determine the number of possible roots of the polynomial equation. 4x 3 – x 2 + 36x – 9 = 0 is a third-degree polynomial. Therefore, this equation will have either 3 real roots or 1 real root and 2 complex roots. 11 2.3.4: The Rational Root Theorem

12 Guided Practice: Example 2, continued 3.Test each possible rational root to find one root. Begin by selecting either a positive or negative possible rational root. Substitute the chosen possibility into the original polynomial. If the result is 0, the number substituted is a rational root. If the result is not 0, then the chosen number is not a rational root. 12 2.3.4: The Rational Root Theorem

13 Guided Practice: Example 2, continued First, try the possible root 1. 4x 3 – x 2 + 36x – 9Original polynomial 4(1) 3 – (1) 2 + 36(1) – 9 Substitute 1 for x. 4 – 1 + 36 – 9 Simplify. 30 Substituting 1 for x does not result in the polynomial equaling 0; therefore, 1 is not a rational root of the polynomial equation. 13 2.3.4: The Rational Root Theorem

14 Guided Practice: Example 2, continued Next, try the possible root. 4x 3 – x 2 + 36x – 9Original polynomial Substitute for x. Simplify. 14 2.3.4: The Rational Root Theorem

15 Guided Practice: Example 2, continued Substituting for x does not result in the polynomial equaling 0; therefore, is not a rational root of the polynomial equation. Next, try the possible root. 15 2.3.4: The Rational Root Theorem

16 Guided Practice: Example 2, continued 4x 3 – x 2 + 36x – 9Original polynomial Substitute for x. Simplify. 0 Substituting for x does result in the polynomial equaling 0; therefore, is a rational root of the polynomial equation. 16 2.3.4: The Rational Root Theorem

17 Guided Practice: Example 2, continued 4.Use synthetic division to find the roots of the depressed polynomial equation. The divisor is. The depressed polynomial is 4x 2 + 36. Find the roots of 4x 2 + 36 = 0. 17 2.3.4: The Rational Root Theorem

18 Guided Practice: Example 2, continued 4x 2 + 36 = 0Depressed polynomial 4(x 2 + 9) = 0Factor out the greatest common factor, 4. x 2 + 9 = 0Divide both sides by 4. x 2 = –9Subtract 9 from both sides. Take the square root of both sides. Simplify. x = ±3i The roots of the depressed polynomial equation 4x 2 + 36 = 0 are ±3i. 18 2.3.4: The Rational Root Theorem

19 Guided Practice: Example 2, continued 5.List the roots of the original polynomial equation. The roots of the equation 4x 3 – x 2 + 36x – 9 = 0 are, 3i, and –3i. 19 2.3.4: The Rational Root Theorem ✔

20 Guided Practice: Example 2, continued 20 2.3.4: The Rational Root Theorem

21 Guided Practice Example 3 Find a third-degree polynomial with rational coefficients that has the roots 6 and 3 – i. 21 2.3.4: The Rational Root Theorem

22 Guided Practice: Example 3, continued 1.Use the Imaginary Root Theorem to find the remaining root. 3 – i is one root. The complex conjugate of 3 – i is 3 + i. 22 2.3.4: The Rational Root Theorem

23 Guided Practice: Example 3, continued 2.Write the polynomial expression. The factors of the polynomial are (x – 6), x – (3 – i), and x – (3 + i). (x – 6)[x – (3 – i)][x – (3 + i)] Write the polynomial in factored form. (x – 6)[x 2 – x(3 + i ) – x(3 – i ) Distribute. + (3 – i )(3 + i)] (x – 6)(x 2 – 3x – ix – 3x + Simplify. ix + 9 + 3i – 3i – i 2 ) 23 2.3.4: The Rational Root Theorem

24 Guided Practice: Example 3, continued (x – 6)[x 2 – 6x + 9 – (–1)]Replace i 2 with –1. (x – 6)(x 2 – 6x + 10) Simplify. x 3 – 12x 2 + 46x – 60 Distribute. x 3 – 12x 2 + 46x – 60 is a third-degree polynomial with rational coefficients and the roots 6 and 3 – i. 24 2.3.4: The Rational Root Theorem ✔

25 Guided Practice: Example 3, continued 25 2.3.4: The Rational Root Theorem

Download ppt "Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this."

Similar presentations

7.5 – Rationalizing the Denominator of Radicals Expressions

7.5 – Rationalizing the Denominator of Radicals Expressions
4.5 Complex Numbers Objectives:

4.5 Complex Numbers Objectives:
Splash Screen.

Splash Screen.
Splash Screen.

Splash Screen.
Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic.

Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic.
Introduction A trinomial of the form that can be written as the square of a binomial is called a perfect square trinomial. We can solve quadratic equations.

Introduction A trinomial of the form that can be written as the square of a binomial is called a perfect square trinomial. We can solve quadratic equations.
Splash Screen.

Splash Screen.
Section 7.8 Complex Numbers  The imaginary number i  Simplifying square roots of negative numbers  Complex Numbers, and their Form  The Arithmetic.

Section 7.8 Complex Numbers  The imaginary number i  Simplifying square roots of negative numbers  Complex Numbers, and their Form  The Arithmetic.
6.2 – Simplified Form for Radicals

6.2 – Simplified Form for Radicals
Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To.

Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To.
Pre-Calculus For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)

Pre-Calculus For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)
Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.

Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.
Solving Polynomial Equations. Fundamental Theorem of Algebra Every polynomial equation of degree n has n roots!

Solving Polynomial Equations. Fundamental Theorem of Algebra Every polynomial equation of degree n has n roots!
Lesson 1-5 The Complex Numbers. Objective: Objective: To add, subtract, multiply, and divide complex numbers.

Lesson 1-5 The Complex Numbers. Objective: Objective: To add, subtract, multiply, and divide complex numbers.
The Rational Zero Theorem

The Rational Zero Theorem
The Fundamental Theorem of Algebra And Zeros of Polynomials

The Fundamental Theorem of Algebra And Zeros of Polynomials
Zeros of Polynomials PolynomialType of Coefficient 5x 3 + 3x 2 + (2 + 4i) + icomplex 5x 3 + 3x 2 + √2x – πreal 5x 3 + 3x 2 + ½ x – ⅜rational 5x 3 + 3x.

Zeros of Polynomials PolynomialType of Coefficient 5x 3 + 3x 2 + (2 + 4i) + icomplex 5x 3 + 3x 2 + √2x – πreal 5x 3 + 3x 2 + ½ x – ⅜rational 5x 3 + 3x.
Introduction Identities are commonly used to solve many different types of mathematics problems. In fact, you have already used them to solve real-world.

Introduction Identities are commonly used to solve many different types of mathematics problems. In fact, you have already used them to solve real-world.
Chapter 5 Polynomials and Polynomial Functions © Tentinger.

Chapter 5 Polynomials and Polynomial Functions © Tentinger.
Introduction Polynomials, or expressions that contain variables, numbers, or combinations of variables and numbers, can be added and subtracted like real.

Introduction Polynomials, or expressions that contain variables, numbers, or combinations of variables and numbers, can be added and subtracted like real.

Similar presentations

Ads by Google