The Factor Theorem.

Slides:



Advertisements
Similar presentations
#1 Factor Each (to prime factors): #2 #3 #4 Solve:
Advertisements

The factor theorem The Factor Theorem states that if f(a) = 0 for a polynomial then (x- a) is a factor of the polynomial f(x). Example f(x) = x 2 + x -
Section 5.5 – The Real Zeros of a Rational Function
3.2 Polynomials-- Properties of Division Leading to Synthetic Division.
Surds and Quadratics AS
Bell Work: Find the values of all the unknowns: R T = R T T + T = 60 R = 3 R =
Whiteboardmaths.com © 2008 All rights reserved
Solving Quadratic Equations by Factoring. Solution by factoring Example 1 Find the roots of each quadratic by factoring. factoring a) x² − 3x + 2 b) x².
Roots & Zeros of Polynomials III
The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:
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.
Polynomials Expressions like 3x 4 + 2x 3 – 6x and m 6 – 4m 2 +3 are called polynomials. (5x – 2)(2x+3) is also a polynomial as it can be written.
7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be.
Copyright © 2009 Pearson Education, Inc. CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions.
1 What we will learn today…  How to divide polynomials and relate the result to the remainder and factor theorems  How to use polynomial division.
 PERFORM LONG DIVISION WITH POLYNOMIALS AND DETERMINE WHETHER ONE POLYNOMIAL IS A FACTOR OF ANOTHER.  USE SYNTHETIC DIVISION TO DIVIDE A POLYNOMIAL BY.
Section 3.3 Real Zeros of Polynomial Functions. Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational.
Factors, Remainders, and Roots, Oh My! 1 November 2010.
Calculus 3.4 Manipulate real and complex numbers and solve equations AS
Roots of Polynomials Quadratics If the roots of the quadratic equation are  and  then the factorised equation is : (x –  )(x –  ) = 0 (x –  )(x –
Function Inverse Quick review. {(2, 3), (5, 0), (-2, 4), (3, 3)} Domain & Range = ? Inverse = ? D = {2, 5, -2, 3} R = {3, 0, 4}
Topic: U4L5 Remainder and Factor Theorems EQ: Can I correctly apply the Remainder and Factor Theorems to help me factor higher order polynomials?
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 4.3 Polynomial Division; The Remainder and Factor Theorems  Perform long division.
Lesson 4-Remainder Theorem 17 December, 2015ML4 MH Objectives : - The remainder and Factor theorems - It’s used to help factorise Polynomials - It’s used.
Section 5.5 The Real Zeros of a Polynomial Function.
Section 5.3(d) Synthetic Substitution. Long division Synthetic Division can be used to find the value of a function. This process is called Synthetic.
3.6 Day 2 Why Synthetic Division? What use is this method, besides the obvious saving of time and paper?
The Remainder Theorem A-APR 2 Explain how to solve a polynomial by factoring.
If a polynomial f(x) is divided by (x-a), the remainder (a constant) is the value of the function when x is equal to a, i.e. f(a). Therefore, we can use.
Theorems About Roots of Polynomial Equations. Find all zeros: f(x)= x +x –x Synthetic Division one zero…need 2 more use (x – k), where.
Solving Polynomials. What does it mean to solve an equation?
Solving equations with polynomials – part 2. n² -7n -30 = 0 ( )( )n n 1 · 30 2 · 15 3 · 10 5 · n + 3 = 0 n – 10 = n = -3n = 10 =
Roots & Zeros of Polynomials III Using the Rational Root Theorem to Predict the Rational Roots of a Polynomial Created by K. Chiodo, HCPS.
7.6 Rational Zero Theorem Objectives: 1. Identify the possible rational zeros of a polynomial function. 2. Find all the rational zeros of a polynomial.
Remainder and Factor Theorems
Chapter 1 Review C Simplifying Algebraic Fractions.
©thevisualclassroom.com To solve equations of degree 2, we can use factoring or use the quadratic formula. For equations of higher degree, we can use the.
Rewrite the numbers so they have the same bases i.e. 8 2 = (2 3 ) 2.
Polynomial and Synthetic Division Objective: To solve polynomial equations by long division and synthetic division.
Section 4.3 Polynomial Division; The Remainder and Factor Theorems Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Solving Polynomials.
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.
Dividing Polynomials Two options: Long Division Synthetic Division.
Remainder and Factor Theorem
Divide by x - 1 Synthetic Division: a much faster way!
When given a root and when not given a root
The Quadratic Formula..
Quadratic Formula Solving for X Solving for quadratic equations.
Remainder Theorem What’s left over?.
Factor Theorem.
Functions Learning Objectives To understand function notation
Quadratic Equations.
The Fundamental Theorem of Algebra
Polynomial Division; The Remainder Theorem and Factor Theorem
f(x) = a(x + b)2 + c e.g. use coefficients to
Lesson 13 – Working with Polynomial Equations
Solving Systems of Equations using Substitution
1.2/1.3 Limits Grand Teton National Park, Wyoming.
Opener Perform the indicated operation.
ALGEBRA II ALGEBRA II HONORS/GIFTED - SECTION 5-2 (Polynomials, Linear Functions, and Zeros) 2/17/2019 ALGEBRA II SECTION.
Roots & Zeros of Polynomials III
4-3: Remainder and Factor Theorems
The Quadratic Formula..
Solving Quadratic Equations by Factorisation
The Quadratic Formula..
Recap from last lesson. On your whiteboards: Fill in the table for
Polynomials Thursday, 31 October 2019.
Function Notation.
Factorisation of Polynomials
Presentation transcript:

The Factor Theorem

What does it do? It helps you factorise polynomials of high order….. like 3!

Function notation, f(x) f(x) stands for “a function of x” This basically says “a polynomial using the letter x” So f(x) could be: f(x) = x3 + 2x2 – x – 2 It saves you having to write out the polynomial every time you refer to it. Also……

Function notation We can use this notation when we substitute values into the polynomial: f(x) = x3 + 2x2 – x – 2 f(1) is just what you get when you substitute 1 into the polynomial f(-2) is what you get when you substitute -2 into the polynomial What is f(3)?

Let’s start simple Factorise x2 – 5x – 6 (x – 6)(x + 1) Now solve x2 – 5x – 6 = 0 (x – 6)(x + 1) = 0 So either x – 6 = 0 or x + 1 = 0 x = 6 x = -1

f(x) = x2 – 5x – 6 What are the values of f(6) and f(-1) (i.e. what do you get when you substitute the values 6 and -1 into the polynomial?) Why?

A little bit harder… x3 + 2x2 – x – 2 What is f(-2) ? What happens when you divide this polynomial by (x + 2)?

Can we see a pattern yet? x3 + 2x2 – x – 2 What happens when you divide this polynomial by (x – 1)? What do you think the value of f(1) will be?

The Factor Theorem If (x – a) is a factor of a polynomial f(x), then: x = a is a solution (root) of the equation f(x) = 0. Conversely, if f(a) = 0, then (x – a) is a factor of f(x)

Using the theorem f(x) = x3 – x2 – 4x + 4 What are f(-2), f(-1), f(0), f(1), f(2)? What are the factors of f(x)?