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5.2 Polynomials, Linear Factors, and Zeros

5.2 Polynomials, Linear Factors, and Zeros. Goals for class: Analyze the factored form of a polynomial Write a polynomial function from its zeros What effect multiple zeros have on a graph Finding a relative Min and max. I. Writing a polynomial in factored form.

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5.2 Polynomials, Linear Factors, and Zeros

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  1. 5.2 Polynomials, Linear Factors, and Zeros Goals for class: Analyze the factored form of a polynomial Write a polynomial function from its zeros What effect multiple zeros have on a graph Finding a relative Min and max

  2. I. Writing a polynomial in factored form Remember: Steps for Factoring 1. GCF is ALWAYS first Then, look for Special Patterns Factors of c that sum to b, if a = 1 or Factors of ac that sum to b x = 0, x = 5, x = -3 2. If we had to solve these two equations, what would the solutions AKA x-intercepts AKA zeros AKA roots AKA and answers be? x = 0, x = -1/2

  3. II. Finding zeros of a polynomial, and sketching the graph 3. These are the zeros of our polynomial Now we need to find the end behavior between each interval We can pick any point between our zeros to see if the graph is above or below the x-axis. We also need to find out what our graph does as x goes to the left and right.

  4. Try: Factor the polynomial and find the zeros, plot the zeros, then sketch the graph.All without a calculator!!! 4.

  5. homework • Page 682#7-25 odd

  6. III. Writing a Polynomial from its zeros 5. 6. Now what is the difference in there graphs if they each only intersect the x-axis three times???

  7. IV. Multiple zeros, multiplicity, and how multiple zeros affect the graph In example 6, we had multiple zeros, since x = 1 showed up twice. Therefore, 1 has a multiplicity of 2, since it shows up two times. If a number shows up as a solution three times, then it would have multiplicity of 3. The multiplicity of a zero effects the shape of the graph. Example 5 had three zeros that all had multiplicity of 1, meaning that the graph looks linear through those there points. Example 6 had two zeros with multiplicity of 1 (3 and 4) and 1 with multiplicity of 2 (x = 1). The graph looks quadratic through at x =1.

  8. Try: Find the zeros, state the multiplicity of each, then sketch the graph 7. X = 0 has multiplicity of 1 which means it will be linear X = 2 has multiplicity of 2 which means it will be quadratic

  9. Challenge:8. Graph

  10. V. Identifying a Relative Minimum and Maximum If the graph of a polynomial function has several turning points, then it may have a relative minimum or a relative maximum.

  11. Find the relative Min and max of the following: 10. 9.

  12. vi. Using a polynomial function in real life2 Minutes to solve

  13. vi. Using a polynomial function in real lifeWith a Partner – 4 minutes - Challenge

  14. With a partner: Page #1-6 (5 minutes to complete, starting NOW)

  15. Homework • Pg 683 #27-39 odd, 40-42, 44, 47-54

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