Lesson 12 Factoring Polynomials Pre Calculus Santowski 1172022

Lesson 12 – Factoring Polynomials Pre. Calculus - Santowski 1/17/2022 Pre. Calculus - Santowski 1

Fast Five n Using technology, graph f(x) = 3 x 3 + x 2 22 x - 24. Sketch & include the max/min points, and intervals of increase and decrease. 1/17/2022 Pre. Calculus - Santowski 2

Lesson Objectives n Use the remainder and rational root theorems and to factor polynomials n Mastery of the factoring of polynomials using the algebraic processes n Reinforce the understanding of the connection between factors and roots n Sketch accurate graphs of polynomial functions 1/17/2022 Pre. Calculus - Santowski 3

(A) Factoring Polynomials – The Remainder Theorem n 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) n PROVE WHY THIS IS TRUE ? !? !? 1/17/2022 Pre. Calculus - Santowski 4

(B) Factoring Polynomials – the Rational Root Theorem n The Rational Root theorem: n Given P(x) = anxn + an-1 xn-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 an 1/17/2022 Pre. Calculus - Santowski 5

(C) Factoring Polynomials – the Rational Root Theorem n Ex 1. To factor P(x) = 2 x – 9 x + 7 x + 6, what values of x Examples 3 2 could you test according to the RRT n Ex 2. To factor P(x) = 3 x 3 – 7 x 2 + 8 x – 2 what values of x could you test according to the RRT n Ex 2. To factor P(x) = 4 x 3 – x 2 + 2 x – 8 what values of x could you test according to the RRT n Ex 2. To factor P(x) = 9 x 4 – x 3 + x – 15 what values of x could you test according to the RRT 1/17/2022 Pre. Calculus - Santowski 6

(D) Factoring Polynomials – The Remainder Theorem – Examples (The Basics) n To factor the following polynomials using the Remainder Theorem what values of x could you test according to the RRT? n Now test your conjectures n P(x) = -x 3 + 7 x – 6 P(x) = x 3 – 5 x 2 – 2 x + 24 P(x) = 2 x 3 – 3 x 2 – 3 x + 2 P(x) = x 4 – x 3 – 3 x 2 + x + 2 n n n 1/17/2022 Pre. Calculus - Santowski 7

(E) Factoring Polynomials – Practice – DAY 2 n Factor g(x) = x 3 + 2 x 2 – 16 x – 32 Factor y = x 3 – 9 x 2 + 24 x – 16 Factor f(x) = x 3 – 6 x 2 + 12 x – 8 n Factor g(x) = -x 3 – 2 x 2 + 16 x + 32 n n 1/17/2022 Math 2 Honors - Santowski 8

(E) Factoring Polynomials – Practice n You are given the graph of y = 2 x 3 + 4 x 2 – 3 x – 6. Factor the polynomial and determine all roots 1/17/2022 Math 2 Honors - Santowski 9

(E) Factoring Polynomials – Practice n For the following polynomials, factor the polynomial, solve for the zeroes and then write the equation as a product of linear factors n P(x) = x 3 - 3 x 2 - 2 x + 6 P(x) = x 3 – 4 x 2 – x + 10 y = x 3 + 4 x 2 + 7 x + 6 n n 1/17/2022 Math 2 Honors - Santowski 10

(E) Factoring Polynomials – Practice You are given the graph of y = 4 x + 4 x – 29 x – 51 x – 18. 4 n 3 2 Factor the polynomial and determine all roots 1/17/2022 Math 2 Honors - Santowski 11

(E) Factoring Polynomials – Practice n Working with Quartic Polynomials: n Factor P(x) = x 4 – x 3 – 3 x 2 + x + 2 Factor f(x) = x 4 + x 3 – 11 x 2 – 9 x + 18 Factor g(x) = x 4 – 3 x 3 + 6 x 2 – 2 x – 12 n n n For these polynomials, factor the polynomial, solve for the zeroes and then write the equation as a product of linear factors 1/17/2022 Math 2 Honors - Santowski 12

(E) Factoring Polynomials – The Remainder Theorem - Examples n Factor P(x) = 2 x 3 + x 2 – 25 x + 12, making use of the RRT and the Remainder Theorem n Now factor P(x) = -2 x 3 – x 2 + 25 x – 12, making use your work in Ex 1 n If x = 4 is root of P(x) = 4 x 3 – 12 x 2 – 19 x + 12, determine the other x-intercepts of P(x) 1/17/2022 Math 2 Honors - Santowski 13

(E) Factoring Polynomials – The Remainder Theorem - Examples n If x = 4 is root of P(x) = 4 x 3 – 12 x 2 – 19 x + 12, determine the other x-intercepts of P(x) 1/17/2022 Math 2 Honors - Santowski 14

(E) Factoring Polynomials – The Remainder Theorem - Examples n n You are given the polynomial: P(x) = 12 x 4 + 32 x 3 – 15 x 2 – 8 x + 3, n And you know that x + 3 is a factor of P(x) and that x = ½ is a zero of P(x). n Find the other zeroes of P(x) 1/17/2022 Math 2 Honors - Santowski 15

(E) Factoring Polynomials – the Rational Root Theorem Examples n SYNTHESIS QUESTION: n WITHOUT USING TECHNOLOGY, graph f(x) = 3 x 3 + x 2 - 22 x - 24 using intercepts, points, and end behaviour. Approximate turning points, max/min points, and intervals of increase and decrease. 1/17/2022 Pre. Calculus - Santowski 16

Homework n Homework: n From the textbook Precalculus with Limits – A Graphing Approach (4 th ed) by Larson, Hostetler & Edwards; Sec 2. 3, p 123 -125, Q 3, 17, 23, 25, 31, 37, 41, 48, 54; APP 81; TIPS 87 1/17/2022 Pre. Calculus - Santowski 17