Lesson 2-6 Solving Polynomial Equations by Factoring – Part 2.

Slides:



Advertisements
Similar presentations
Remainder and Factor Theorems
Advertisements

Chapter 6: Polynomials and Polynomial Functions Section 6
Lesson 2.6 Pre-Calc Part 2 When trying to ‘factor’ a quadratic into two binomials, we only ever concern ourselves with the factors of the ‘a’ --- leading.
6.5 & 6.6 Theorems About Roots and the Fundamental Theorem of Algebra
4.4 Notes The Rational Root Theorem. 4.4 Notes To solve a polynomial equation, begin by getting the equation in standard form set equal to zero. Then.
Pre Calc Lesson 2.2 Synthetic Division ‘Remainder’ and ‘Factor’ Theorems Review Long Division: 5365 ÷ 27 Now review ‘long division’ of polynomials: (2x.
Solving Polynomial Equations. Fundamental Theorem of Algebra Every polynomial equation of degree n has n roots!
The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division,
Remainder and Factor Theorem Unit 11. Definitions Roots and Zeros: The real number, r, is a zero of f(x) iff: 1.) r is a solution, or root of f(x)=0 2.)
General Results for Polynomial Equations
The Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all.
Finding the Potential Zeros.  A Theorem that provides a complete list of possible Rational Roots or Zeroes of the Polynomial Equation.  A Root or Zero.
Warm up Use Synthetic Division: 1. 3x 2 – 11x + 5 x – x 5 + 3x 3 +1 x + 2.
2.3 Synthetic Substitution OBJ:  To evaluate a polynomial for given values of its variables using synthetic substitution.
Factor Theorem & Rational Root Theorem
7.5.1 Zeros of Polynomial Functions
Using Technology to Approximate Roots of Polynomial Equations.
Quick Crisp Review Zeros of a polynomial function are where the x-intercepts or solutions when you set the equation equal to zero. Synthetic and long division.
7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
5.5 Theorems about Roots of Polynomial Equations P
Warm - Up Find the Vertex of f(x) = x 2 – 2x + 4.
The Remainder and Factor Theorems. Solve by Using Long Division Example 1Example 2.
Lesson 2.5, page 312 Zeros of Polynomial Functions Objective: To find a polynomial with specified zeros, rational zeros, and other zeros, and to use Descartes’
Zeros of Polynomials 2.5.
Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,
The Original f(x)=x 3 -9x 2 +6x+16 State the leading coefficient and the last coefficient Record all factors of both coefficients According to the Fundamental.
Solving Polynomials. What does it mean to solve an equation?
Solving Polynomial Equations by Factoring Factoring by grouping Ex. 1. Solve:
Solving Equations Binomials Simplifying Polynomials
7.6 Rational Zero Theorem Depressed equation. All the possible rational Zeros To find all the possible rational zero, take all the factors of the last.
6.5 Theorems About Roots of Polynomial Equations
6.5 Day 1 Rational Zeros Theorem. If is in simplest form and is a rational root of the polynomial equation With integer coefficients, then p must be a.
Rational Root Theorem Definitions Steps Examples.
LESSON 5.6 Rational Zeros of Polynomial Functions.
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.
Lesson 11-2 Remainder & Factor Theorems Objectives Students will: Use synthetic division and the remainder theorem to find P(r) Determine whether a given.
PreCalculus Section 2.6 Solve polynomial equations by factoring and the Rational Roots Theorem. Solve by factoring: x 3 + 5x 2 – 4x – 20 = 0 x 6 – x 3.
Thurs 12/3 Lesson 5 – 5 Learning Objective: To write equations given roots Hw: Pg. 315#23 – 29 odd, 37 – 41 odd, 42, 45.
Solving polynomial equations
Solving Polynomials. Factoring Options 1.GCF Factoring (take-out a common term) 2.Sum or Difference of Cubes 3.Factor by Grouping 4.U Substitution 5.Polynomial.
Polynomials. DegreeNameExample 0Constant 1Linear 2Quadratic 3Cubic 4Quartic 5Quintic Some of the Special Names of the Polynomials of the first few degrees:
Polynomial Long Division
STD 3: To divide polynomials using long division and synthetic division and the rational root theorem Warm Up /15.
3.2 Division of Polynomials. Remember this? Synthetic Division 1. The divisor must be a binomial. 2. The divisor must be linear (degree = 1) 3. The.
3.3 Real Zeros of Polynomials. Rational Zero Theorem If the polynomial P(x) has integer coefficients then every rational zero of P is of the form p/q.
Dividing Polynomials Two options: Long Division Synthetic Division.
Polynomial Long Division Review
Section 6.5 Theorems about Roots of Polynomial Equations Objective: Students will be able to solve equations using Theorems involving roots. Rational Root.
Polynomial Long Division Review
Factor Theorem & Rational Root Theorem
PreCalculus Section 2.6 Solve polynomial equations by factoring and the Rational Roots Theorem. Solve by factoring: x3 + 5x2 – 4x – 20 = 0 x6 – x3 – 6.
3.3 Real Zeros of Polynomials
2.5 Zeros of Polynomial Functions
Real Zeros Intro - Chapter 4.2.
Rational Zero Theorem Rational Zero Th’m: If the polynomial
7.5 Zeros of Polynomial Functions
5-5 Theorems About Roots of Polynomial Equations
7.5 Zeros of Polynomial Functions
Factor Theorem & Rational Root Theorem
Finding polynomial roots
Finding Zeros of Polynomials
Warm-up Complete this as a group on you Board. You have 15 minutes
Factor Theorem & Rational Root Theorem
Finding Zeros of Polynomials
Factor Theorem & Rational Root Theorem
4.3 – The Remainder and Factor Theorems
8-5 Rational Zero Theorem
21 = P(x) = Q(x) . (x - r) + P(r) 4.3 The Remainder Theorem
2.5 Apply the Remainder and Factor Theorem
Warm Up.
Presentation transcript:

Lesson 2-6 Solving Polynomial Equations by Factoring – Part 2

Objective:

To solve polynomial equations by various methods of factoring, including the use of the rational root theorem. Objective:

When trying to factor a quadratic into two binomials, we only ever concern ourselves with the factors of the a (leading coefficient) and c (constant term).

Solve:

3x 2 – 11x – 4 = 0

Solve: 3x 2 – 11x – 4 = 0 (3x + 1)(x – 4) = 0 Solving for x  x = - 1/3 or x = 4

Solve: So we only concerned ourselves with the factors of 3 and 4. 3x 2 – 11x – 4 = 0 (3x + 1)(x – 4) = 0 Solving for x  x = - 1/3 or x = 4

We call the possible factors of c  p values.

We call the possible factors of c  p values. We call the possible factors of a  q values.

This leads us into what is called the Rational Roots Theorem.

This leads us into what is called the Rational Roots Theorem. Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term.

This leads us into what is called the Rational Roots Theorem. Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term. P(x) = a n x n + a n-1 x n-1 + …+ a 0 where a 0 ≠0

This leads us into what is called the Rational Roots Theorem. If one of the roots of the equation P(x) = 0 is x = p/q where p and q are nonzero integers with no common factor other than 1, then p must be a factor of a 0 and q must be a factor of a n ! P(x) = a n x n + a n-1 x n-1 + …+ a 0 where a 0 ≠0

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0

Note: If there are any rational roots, then they must be in the form of p/q.

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Note: If there are any rational roots, then they must be in the form of p/q. 1 st : List all possible q values: ±1(±3)

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Note: If there are any rational roots, then they must be in the form of p/q. 1 st : List all possible q values: ±1(±3) 2 nd : List all possible p values: ±1(±4); (±2)(±2)

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Therefore, if there is a rational root then it must come from this list of possible p/q values:

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Therefore, if there is a rational root then it must come from this list of possible p/q values: p/q  ±(1/1, 1/3, 4/1, 4/3, 2/1, 2/3) which means there are 12 possibilities!

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Now, determine whether any of the possible rational roots are really roots. If so, then find them.

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Lets first evaluate x = 1.

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Lets first evaluate x = 1. Do you remember the quick and easy way to see if x = 1 is a root?

According to the rational roots theorem what are the possible rational roots of : Px) = 3x x x 2 – 4 = 0 Now, check the other possibilities using synthetic division.

Pg – 39 odd Assignment: