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Lesson 12 – Factoring Polynomials

Lesson 12 – Factoring Polynomials. PreCalculus - Santowski. Fast Five. Using technology, graph f(x) = 3x 3 + x 2 - 22x - 24. Sketch & include the max/min points, and intervals of increase and decrease. Lesson Objectives.

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Lesson 12 – Factoring Polynomials

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  1. Lesson 12 – Factoring Polynomials PreCalculus - Santowski PreCalculus - Santowski

  2. Fast Five • Using technology, graph f(x) = 3x3 + x2 - 22x - 24. Sketch & include the max/min points, and intervals of increase and decrease. PreCalculus - Santowski

  3. 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 PreCalculus - Santowski

  4. (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 ?!?!?!?!? PreCalculus - Santowski

  5. (B) Factoring Polynomials – the Rational Root Theorem • The Rational Root theorem: • Given P(x) = anxn + an-1xn-1 + ….. + a1x1 + a0, 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 a0 and b must be a divisor of an PreCalculus - Santowski

  6. (C) Factoring Polynomials – the Rational Root Theorem - Examples • Ex 1. To factor P(x) = 2x3 – 9x2 + 7x + 6, what values of x could you test according to the RRT • Ex 2. To factor P(x) = 3x3 – 7x2 + 8x – 2 what values of x could you test according to the RRT • Ex 2. To factor P(x) = 4x3 – x2 + 2x – 8 what values of x could you test according to the RRT • Ex 2. To factor P(x) = 9x4 – x3 + x – 15 what values of x could you test according to the RRT PreCalculus - Santowski

  7. (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) = -x3 + 7x – 6 • P(x) = x3 – 5x2 – 2x + 24 • P(x) = 2x3 – 3x2 – 3x + 2 • P(x) = x4 – x3 – 3x2 + x + 2 PreCalculus - Santowski

  8. (E) Factoring Polynomials – Practice – DAY 2 • Factor g(x) = x 3 + 2x 2 – 16x – 32 • Factor y = x3 – 9x2 + 24x – 16 • Factor f(x) = x3 – 6x2 + 12x – 8 • Factor g(x) = -x 3 – 2x 2 + 16x + 32 Math 2 Honors - Santowski

  9. (E) Factoring Polynomials – Practice • You are given the graph of y = 2x3 + 4x2 – 3x – 6. Factor the polynomial and determine all roots Math 2 Honors - Santowski

  10. (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) = x3 - 3x2 - 2x + 6 • P(x) = x3 – 4x2 – x + 10 • y = x3 + 4x2 + 7x + 6 Math 2 Honors - Santowski

  11. (E) Factoring Polynomials – Practice • You are given the graph of y = 4x4 + 4x3 – 29x2 – 51x – 18. Factor the polynomial and determine all roots Math 2 Honors - Santowski

  12. (E) Factoring Polynomials – Practice • Working with Quartic Polynomials: • Factor P(x) = x4 – x3 – 3x2 + x + 2 • Factor f(x) = x4 + x3 – 11x2 – 9x + 18 • Factor g(x) = x4 – 3x3 + 6x2 – 2x – 12 • For these polynomials, factor the polynomial, solve for the zeroes and then write the equation as a product of linear factors Math 2 Honors - Santowski

  13. (E) Factoring Polynomials – The Remainder Theorem - Examples • Factor P(x) = 2x3 + x2 – 25x + 12, making use of the RRT and the Remainder Theorem • Now factor P(x) = -2x3 – x2 + 25x – 12, making use your work in Ex 1 • If x = 4 is root of P(x) = 4x3 – 12x2 – 19x + 12, determine the other x-intercepts of P(x) Math 2 Honors - Santowski

  14. (E) Factoring Polynomials – The Remainder Theorem - Examples • If x = 4 is root of P(x) = 4x3 – 12x2 – 19x + 12, determine the other x-intercepts of P(x) Math 2 Honors - Santowski

  15. (E) Factoring Polynomials – The Remainder Theorem - Examples • You are given the polynomial: • P(x) = 12x4 + 32x3 – 15x2 – 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) Math 2 Honors - Santowski

  16. (E) Factoring Polynomials – the Rational Root Theorem - Examples • SYNTHESIS QUESTION: • WITHOUT USING TECHNOLOGY, graph f(x) = 3x3 + x2 - 22x - 24 using intercepts, points, and end behaviour. Approximate turning points, max/min points, and intervals of increase and decrease. PreCalculus - Santowski

  17. Homework • Homework: • From the textbook Precalculus with Limits – A Graphing Approach (4th ed) by Larson, Hostetler & Edwards; Sec 2.3, p123-125, Q3,13,17,23,25,31,37,41,48,54; APP 81; TIPS87 PreCalculus - Santowski

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