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MODULE 5 • TOPIC 2 • LESSON 3: SOLVING LINEAR EQUATIONS

Objectives

  • Describe similarities between polynomials and integers.

  • Determine factors of a polynomial using one or more roots of the polynomial.

  • Compare polynomial long division to integer long division.

  • Determine factors through polynomial long division.

  • Use the remainder theorem to evaluate polynomial equations and functions.

New Key Terms

  • Factor Theorem: A theorem that states if a polynomial function p(x) has a root at x = r, then (x - r) is a factor of p(x).

  • Polynomial Long Division: The process of dividing one polynomial by another of equal or lesser degree, similar to how long division is done with integers.

  • Remainder Theorem: When a polynomial function f(x) is divided by a linear polynomial (x - r), the remainder is equal to f(r).

Introduction to Polynomial Division

  • Familiarity with integer long division aids in understanding polynomial long division.

  • The polynomial long division algorithm can be applied to find both the quotient polynomial and the remainder polynomial.

Setting the Stage

  • Communicate core objectives and key terms.

  • Activate prior learning by engaging with narrative content.

  • Provide direction through the Essential Question.

The x - r Factor

  1. Analyzing the Function:

    • Given function h(x) = 2x² + 5x - 12

    • Characteristics of h(x):

      • Quadratic function, upward-opening parabola, absolute minimum vertex, axis of symmetry is x = -5/4.

      • Y-intercept at (0, -12), x-intercepts at (-4, 0).

    • Zeros of h(x):

      • The function can have 0, 1, or 2 real zeros. In this case, h(x) has 2 zeros.

      • Factor Theorem: A polynomial function p(x) has x - r as a factor if p(r) = 0.

Using the Factor Theorem

  1. Analysis of Factor:

    • Consider d(x) = (x + 4) and the product d(x) · q(x) = h(x).

    • q(x) must be linear as the product of two linear factors yields a quadratic function.

    • The zero of q(x) estimated between 1 and 2 leads to finding q(x).

    • Use trial and error or graph estimation to deduce potential linear factors.

  2. Worked Example (h(x) and factoring):

    • Given h(-4):

      • h(-4) = 2(-4)² + 5(-4) - 12

      • = 32 - 20 - 12

      • = 0

    • This demonstrates (x + 4) is a linear factor of the polynomial.

The Fundamental Theorem of Algebra

  • Every polynomial of degree n must have n roots and can be expressed as the product of n linear factors in the form (ax + b).

  • Example: The polynomial 2x² - 3x - 9 can be factored into (2x + 3)(x - 3).

Polynomial Long Division Algorithm

  1. Definition:

    • a ÷ b results in a quotient and a remainder, where b ≠ 0.

  2. Example Comparison (Integer vs. Polynomial Long Division):

    • Integer: 3660 ÷ 12 = 305 with a remainder of 0.

    • Polynomial: (2x² + 5x - 12) ÷ (x + 4) results in the step-by-step algorithm applied to both integer and polynomial forms showing parallels.

  3. Step-by-Step Polynomial Long Division:

    • A. Divide leading terms: 2x² ÷ x = 2x

    • B. Multiply and subtract from the original polynomial.

    • C. Bring down the next term and repeat until finished, confirming a remainder of 0, indicating (x + 4) is a factor.

  4. Importance of Remainders:

    • A non-zero remainder indicates that the divisor is not a factor of the dividend, while a zero remainder confirms it is a factor.

Utilizing the Remainder Theorem

  1. Application in Evaluating Polynomial Functions:

    • When a polynomial f(x) is divided by (x - r), then the remainder R = f(r).

    • Cases:

      • For example, p(x) = 12x² - 19x + 10 shows (x - 2) is not a factor because p(2) yields a non-zero remainder.

      • In contrast, evaluating x + 18 for the polynomial x² - 324 gives a zero remainder, confirming it as a factor.

  2. Table of Values and Evaluations:

    • Provide cases of polynomials to illustrate remainder evaluations and corresponding zeros.

    • Encourage clarifying student deductions through examples.

Conclusion: Links Between Theorems

  • Factor Theorem: A polynomial has (x - r) as a factor if and only if f(r) = 0.

  • Remainder Theorem: f(r) represents the remainder when f(x) is divided by (x - r).

  • The importance of recognizing the relationship between these theorems and their applications in polynomial division for problem-solving.

Practice Problems and Assignments

  1. Demonstrate understanding of the concepts through various polynomial evaluations using factor and remainder theorems.

  2. Assign exercises requiring polynomial long division to find factors, ensuring a comprehensive grasp of the material covered.