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Understanding Polynomial Operations: Quadratic Formula, Division, and Factor Theorems

In this lesson, students will learn to use the quadratic formula to solve equations and apply synthetic substitution to evaluate expressions at specific values (e.g., for x = 2.3). We will model real-world income scenarios with polynomial functions and utilize the remainder and factor theorems to factor polynomials effectively. The lesson includes long division techniques for numbers and polynomials, emphasizing the significance of accurate steps, such as dividing and bringing down terms. Practice assignments will reinforce these concepts.

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Understanding Polynomial Operations: Quadratic Formula, Division, and Factor Theorems

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  1. Bellwork • Use the quadratic formula to solve • Use synthetic substitution to evaluate for x = 2 3) A company’s income is modeled by the function . What is the value of P when x = 200?

  2. Lesson 5.5 Apply the remainder and factor theorems

  3. Objective • Use theorems to factor polynomials

  4. Standard • 3.0 – Students are adept at operations on polynomials, including long division

  5. Academic Vocabulary • Polynomial long division • Synthetic division

  6. Use Long Division 1) 147 divided by 9 2) 1214 divided by 11

  7. What are the steps for long division? 1) Divide the first term • Multiply • Subtract • Bring down next term • Repeat steps 1 - 4

  8. Polynomial Long Division

  9. Now apply steps to Polynomials

  10. Practice

  11. Practice

  12. Synthetic Division

  13. Example • Remember to use k which is opposite of what you see for the constant of the binomial!

  14. Practice If the remainder is 0 than x + 2 is a factor and the resulting quotient is also a factor which needs to be rewritten as a polynomial Remainder x2 x1 x0 Constant

  15. Now factor remaining polynomial

  16. Practice

  17. The Real World

  18. Homework Part 1 p.366 #1, 2, 3, 5, 11, 19, 25, 35, 41 Part 2 p.366 #7, 13, 17, 23, 27

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