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Do Now

Do Now. Decide whether x-1 is a factor of f(x) Factor f(x) into linear factors given that 2 is a zero of f(x) What is the relationship between zeroes and factors?. Today’s Learning Objectives. Use rational zeroes theorem to compile a list of all possible rational zeroes

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Do Now

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  1. Do Now • Decide whether x-1 is a factor of f(x) • Factor f(x) into linear factors given that 2 is a zero of f(x) • What is the relationship between zeroes and factors?

  2. Today’s Learning Objectives Use rational zeroes theorem to compile a list of all possible rational zeroes Combine rational zeroes theorem with remainder theorem to test which possibilities are actual solutions

  3. This week is the beef • We are adding beefy theorems this week that require us to know how to use remainder theorem as a tool to test for zeroes • Must know relationship between factors and theorems

  4. Rational Zeros Theorem • Gives us a method to determine all possible candidates for rational zeros of a polynomial function • If is a rational number written in lowest terms, and if it is a zero of f, then p is a factor of the constant term and q is a factor of the leading coefficient • Key points: • p/q in lowest terms • p is factor of the constant • q is factor of the leading coefficient

  5. RZT- compiling the list • Use RZT to compile a list of all possible rational zeroes of the following functions:

  6. RZT- what to do with the list? • We have compiled the list of all possible zeros. Now what should we do with it? What should we use it for? • Turn and talk to your neighbor and discuss some possible uses of this list; I will randomly call on a few people to share

  7. RZT- why? • RZT compiles the list of all possibilities • Synthetic division/ remainder theorem narrows down which ones are true • Last week: Test to see if a given value is a zero • This week: Test to see if a possible value is a zero

  8. RZT- ex #1 • List all possible rational zeros of • List all p’s: • List all q’s: • All possible combinations (without repeats) of p/q in lowest terms: • Find all rational zeros and FACTOR f(x) into linear factors. • Use remainder theorem, looking for R=0

  9. RZT- ex #2 • List all possible rational zeros of • List all p’s: • List all q’s: • All possible combinations (without repeats) of p/q in lowest terms: • Find all rational zeros and FACTOR f(x) into linear factors. • Use remainder theorem, looking for R=0

  10. RZT Summary • Use RZT to compile a list of all possibilities • Use Remainder Theorem (with synthetic division) to test possibilities • Seems long… how do we know when to stop? • Number of Zeros Theorem: • A function defined by a polynomial of degree n has at most n distinct zeros • Has as many zeros as its highest exponent

  11. Class work- due at end of hour • Page 338, section 3.3 #35-41 odd (4 problems) • Quiz at beginning of hour tomorrow: one rational zeros theorem question asking you to fully factor a polynomial

  12. 3.3 Continue Homework • We have not finished the section yet • #8-60, multiples of 4 • Today’s material: 36 & 40 even

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