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5.2

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5.2

  1. 1. 5.2 Polynomials, LinearFactors, and Zeros
  2. 2. Writing a Polynomial in FactoredForm In Chapter 4, we solved quadratic functions by factoring and setting each factor equal to zero We can solve some polynomial functions in a similar way. ◦ Remember: ALWAYS factor out the GCF first!
  3. 3. Linear Factors, Roots, Zeros,and x-intercepts The following are equivalent statements about a real number b and a polynomial P(x) ◦ (x – b) is a linear factor of the polynomial P(x) ◦ b is a zero of the polynomial function y = P(x) ◦ b is a root (or solution) of the polynomial equation P(x) = 0 ◦ b is an x-intercept of the graph of y = P(x)
  4. 4. Example: Write each polynomialin factored form. Then, find thezeros of the function.
  5. 5. Example: Write each polynomialin factored form. Then, find thezeros of the function.
  6. 6. Graphing a PolynomialFunction1. Find the zeros and plot them2. Find points between the zeros and plot them3. Determine the end behavior4. Sketch the graph
  7. 7. Example: Find the zeros of thefunction. Then graph thefunction.
  8. 8. Example: Find the zeros of thefunction. Then graph thefunction.
  9. 9. The Factor Theorem The factor theorem describes the relationship between the linear factors of a polynomial and the zeros of a polynomial. The Factor Theorem The expression x – a is a factor of a polynomial if and only if the value a is a zero of the related polynomial function
  10. 10. Using the Factor Theorem toWrite a Polynomial1. Write each zero as a factor2. Multiply and combine like terms
  11. 11. Example: Write a polynomialfunction in standard form with thegiven zerosX = –2, 2, and 3
  12. 12. Example: Write a polynomialfunction in standard form with thegiven zerosX = –2, –2, 2 and 3
  13. 13. Homework P293 #7 – 26, 61 – 63
  14. 14. Multiple Zeros and Multiplicity A multiple zero is a linear factor that is repeated when the polynomial is factored completely The multiplicity of a zero is the number of times the linear factor is repeated in the factored form of the polynomial. ◦ If a zero is of even multiplicity, then the graph touches the x-axis and “turns around” ◦ If a zero is of odd multiplicity, then the graph crosses the x-axis
  15. 15. Example: Find the zeros of thefunction. State the multiplicity ofmultiple zeros. What does the multiplicity tell you about the graph?

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