Lesson 12 – Factoring Polynomials PreCalculus - Santowski 1/5/20161 PreCalculus - Santowski

Fast Five Using technology, graph f(x) = 3x 3 + x x Sketch & include the max/min points, and intervals of increase and decrease. 1/5/20162 PreCalculus - Santowski

Lesson Objectives Use the remainder and rational root theorems and to factor polynomials Mastery of the factoring of polynomials using the algebraic processes Reinforce the understanding of the connection between factors and roots Sketch accurate graphs of polynomial functions 1/5/20163 PreCalculus - Santowski

(A) Factoring Polynomials – The Remainder Theorem the remainder theorem states "when a polynomial, P(x), is divided by (ax - b), and the remainder contains no term in x, then the remainder is equal to P(b/a) PROVE WHY THIS IS TRUE ?!?!?!?!? 1/5/20164 PreCalculus - Santowski

(B) Factoring Polynomials – the Rational Root Theorem The Rational Root theorem: Given P(x) = a n x n + a n-1 x n-1 + ….. + a 1 x 1 + a 0, if P(x) = 0 has a rational root of the form a/b and a/b is in lowest terms, then a must be a divisor of a 0 and b must be a divisor of a n 1/5/20165 PreCalculus - Santowski

(C) Factoring Polynomials – the Rational Root Theorem - Examples Ex 1. To factor P(x) = 2x 3 – 9x 2 + 7x + 6, what values of x could you test according to the RRT Ex 2. To factor P(x) = 3x 3 – 7x 2 + 8x – 2 what values of x could you test according to the RRT Ex 2. To factor P(x) = 4x 3 – x 2 + 2x – 8 what values of x could you test according to the RRT Ex 2. To factor P(x) = 9x 4 – x 3 + x – 15 what values of x could you test according to the RRT 1/5/20166 PreCalculus - Santowski

(D) Factoring Polynomials – The Remainder Theorem – Examples (The Basics) To factor the following polynomials using the Remainder Theorem  what values of x could you test according to the RRT? Now test your conjectures P(x) = -x 3 + 7x – 6 P(x) = x 3 – 5x 2 – 2x + 24 P(x) = 2x 3 – 3x 2 – 3x + 2 P(x) = x 4 – x 3 – 3x 2 + x + 2 1/5/2016 PreCalculus - Santowski 7

(E) Factoring Polynomials – Practice – DAY 2 Factor g(x) = x 3 + 2x 2 – 16x – 32 Factor y = x 3 – 9x x – 16 Factor f(x) = x 3 – 6x x – 8 Factor g(x) = -x 3 – 2x x /5/20168 Math 2 Honors - Santowski

(E) Factoring Polynomials – Practice You are given the graph of y = 2x 3 + 4x 2 – 3x – 6. Factor the polynomial and determine all roots 1/5/2016 Math 2 Honors - Santowski 9

(E) Factoring Polynomials – Practice For the following polynomials, factor the polynomial, solve for the zeroes and then write the equation as a product of linear factors P(x) = x 3 - 3x 2 - 2x + 6 P(x) = x 3 – 4x 2 – x + 10 y = x 3 + 4x 2 + 7x + 6 1/5/ Math 2 Honors - Santowski

(E) Factoring Polynomials – Practice You are given the graph of y = 4x 4 + 4x 3 – 29x 2 – 51x – 18. Factor the polynomial and determine all roots 1/5/2016 Math 2 Honors - Santowski 11

(E) Factoring Polynomials – Practice Working with Quartic Polynomials: Factor P(x) = x 4 – x 3 – 3x 2 + x + 2 Factor f(x) = x 4 + x 3 – 11x 2 – 9x + 18 Factor g(x) = x 4 – 3x 3 + 6x 2 – 2x – 12 For these polynomials, factor the polynomial, solve for the zeroes and then write the equation as a product of linear factors 1/5/ Math 2 Honors - Santowski

(E) Factoring Polynomials – The Remainder Theorem - Examples Factor P(x) = 2x 3 + x 2 – 25x + 12, making use of the RRT and the Remainder Theorem Now factor P(x) = -2x 3 – x x – 12, making use your work in Ex 1 If x = 4 is root of P(x) = 4x 3 – 12x 2 – 19x + 12, determine the other x-intercepts of P(x) 1/5/2016 Math 2 Honors - Santowski 13

(E) Factoring Polynomials – The Remainder Theorem - Examples If x = 4 is root of P(x) = 4x 3 – 12x 2 – 19x + 12, determine the other x-intercepts of P(x) 1/5/2016 Math 2 Honors - Santowski 14

(E) Factoring Polynomials – The Remainder Theorem - Examples You are given the polynomial: P(x) = 12x x 3 – 15x 2 – 8x + 3, And you know that x + 3 is a factor of P(x) and that x = ½ is a zero of P(x). Find the other zeroes of P(x) 1/5/2016 Math 2 Honors - Santowski 15

(E) Factoring Polynomials – the Rational Root Theorem - Examples SYNTHESIS QUESTION: WITHOUT USING TECHNOLOGY, graph f(x) = 3x 3 + x x - 24 using intercepts, points, and end behaviour. Approximate turning points, max/min points, and intervals of increase and decrease. 1/5/ PreCalculus - Santowski

Homework Homework: From the textbook Precalculus with Limits – A Graphing Approach (4 th ed) by Larson, Hostetler & Edwards; Sec 2.3, p , Q3,13,17,23,25,31,37,41,48,54; APP  81; TIPS  87 1/5/2016 PreCalculus - Santowski 17