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Exploring Division of Polynomial Functions: Long and Synthetic Methods

In this lesson, students will investigate how to divide polynomial functions using both long division and synthetic division methods. They will learn about the essential question: How do you divide a polynomial by another polynomial? The lesson includes the exploration of the Remainder Theorem and the Factor Theorem, allowing students to understand important properties of polynomial functions. Homework assignments focus on practical applications to reinforce these concepts. This lesson aligns with standard MM4A4 and involves comparing various types of functions.

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Exploring Division of Polynomial Functions: Long and Synthetic Methods

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  1. Warm Up P 112 # 1-8 (not the vocabulary)

  2. Math IV Lesson 9 Real Zeros of polynomial functions Essential Question: How do you divide a polynomial by another polynomial? Standard: MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

  3. New Vocabulary • Improper fraction: a fraction in which the numerator is larger than the denominator • Proper fraction: a fraction in which the numerator is less than the denominator. • Long division: a process for dividing polynomials • Synthetic division: the shorter process for dividing polynomials when the polynomial you are dividing by has the form x + a.

  4. Long division • Divide 6x3 -19x2 + 16 x – 4 by x – 2 And use the result to factor the polynomial completely.

  5. Synthetic division • Use synthetic division to divide x4 -10x2 -2x + 4 by x + 3

  6. The Remainder theorem If a polynomial f(x) is divided by x – k, the remainder is r = f(k)

  7. The remainder theorem Use synthetic division to find f(-2) when F(x) = 3x3 + 8x2 +5x – 7

  8. The factor theorem A polynomial f(x) has a factor (x – k) if and only if f(k) = 0

  9. Homework • Page 127 # 1-4, 15-18, 35

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