Algebra II Chapter 5: Part 2gruver.weebly.com/uploads/9/8/2/7/9827712/alg_ii... · Alg II Chapter 5...
Transcript of Algebra II Chapter 5: Part 2gruver.weebly.com/uploads/9/8/2/7/9827712/alg_ii... · Alg II Chapter 5...
Alg II Chapter 5 Notes Part 2.notebook
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November 09, 2015
Algebra IIChapter 5:
Polynomials and Polynomial Functions
Part 2
Chapter 5 Lesson 5Apply the Remainder and Factor TheoremsVocabulary
Alg II Chapter 5 Notes Part 2.notebook
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Example 1: Use Polynomial Long Division
Reminder:Since you are using subtraction to find the "remainder", make sure to distribute the negative to all terms being subtracted!
Example 2: Use Polynomial Long Division with a Linear Divisor
Alg II Chapter 5 Notes Part 2.notebook
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Example 3: Use Synthetic DivisionHow to Use Synthetic Division:1. This can only be completed with divisor in the format of x k.2. Make sure to use the solution, not the factored piece! (Change the sign of the k value!)3. Complete like synthetic substitution. Don't forget to add zeroes for the variables that are not there.
Example 4: Factor a Polynomial
Solution!
Factored form!
1. Divide the polynomial by the given factor.2. Factor the result! (Use the factoring methods we discussed before: quadratic form, grouping, special patterns, etc.)3. Don't forget to include the given factor in the factored form.
Alg II Chapter 5 Notes Part 2.notebook
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November 09, 2015
Example 5: Standardized Test Practice
1. Divide the polynomial by the given factor or solution.2. Factor the result! (Use the factoring methods we discussed before: quadratic form, grouping, special patterns, etc.)3. Don't forget to include the given factor in the factored form.4. Set each factored part with a variable equal to zero and solve.5. Check with your calculator!!! (We found zeroes before. Check your graphing sheet if needed!)
Solution!
Factored form!
Example 6: Use a Polynomial Model
Alg II Chapter 5 Notes Part 2.notebook
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Chapter 5 Lesson 6Find Rational ZeroesVocabulary
Reminder:
Zero of a Function
Constant Term
Leading Coefficient
Example 1: List Possible Rational Zeroes
You must consider all possible combinations!
Alg II Chapter 5 Notes Part 2.notebook
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Example 2: Find Zeroes When the Leading Coefficient is 11. List the possible rational zeroes.2. Test the possibilities using synthetic division.3. When one works, write it as a factor and write the resulting polynomial.4. You can test other possible zeroes in the resulting polynomial if it is not factorable yet.5. Fully factor the result.6. Solve.7. Check with your calculator!!!
Find the zeroes.
Example 3: Find Zeroes When the Leading Coefficient is Not 11. List the possible rational zeroes.2. Use your calculator to estimate which possible zeroes might work.3. Test your best estimates. 4. When one works, write it as a factor and divide it out.5. Repeat step 13 until you can easily factor the remaining polynomial.6. Solve.7. Check with your calculator!
Find the zeroes.
Alg II Chapter 5 Notes Part 2.notebook
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Example 4: Solve a MultiStep Problem
Chapter 5 Lesson 7Apply the Fundamental Theorem of AlgebraVocabulary
Reminder:
Irrational Conjugates
Complex Conjugates
Alg II Chapter 5 Notes Part 2.notebook
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Example 1: Find the Number of Solutions or Zeroes
What does that mean?The number of zeroes is equal to the degree of the function! Notes:
Some solutions are repeated!Some solutions are imaginary!!!
Example 2: Find the Zeroes of a Polynomial Function1. List the possible rational zeroes. Consider how many zeroes the polynomial has. 2. Test the possibilities using synthetic division. Sometimes there will be a repeated one!3. When one works, write it as a factor.4. Factor the remaining polynomial.5. Solve for the zeroes.
Alg II Chapter 5 Notes Part 2.notebook
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Example 3: Use Zeroes to Write a Polynomial Function
1. Write each zero as a factor. When dealing with complex or irrational zeroes, remember to include both the positive and negative values.2. Multiply the factors together.
Example 4: Use Descartes' Rule of Signs
Number of Sign Changes:
Number of Sign Changes when x is Negative:
Alg II Chapter 5 Notes Part 2.notebook
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Example 5: Approximate Real ZeroesUsing a graphing calculator to approximate zeroes:1. Touch the y= button.2. Enter the equation.3. Touch the GRAPH button.4. Touch 2nd then CALC.5. Scroll to 2:zero and touch ENTER.6. Scroll to the left side of the graph where it crosses the x axis. Touch ENTER.7. Scroll to the right side of the graph where it crosses the x axis. Touch ENTER.8. Touch ENTER again.
Chapter 5 Lesson 8Analyze Graphs of Polynomial FunctionsVocabulary
Alg II Chapter 5 Notes Part 2.notebook
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Example 1: Use xintercepts to Graph a Polynomial FunctionGraphing a Polynomial Function:1. Find and plot the intercepts.2. Figure out the end behavior.3. Make a table to find and plot points between
and beyond the intercepts.4. Draw the graph.5. Check with your calculator!
State the domain and range.
Domain:
Range:
Example 2: Find Turning Points
Using a graphing calculator to approximate local maximums and minimums:1. Touch the y= button.2. Enter the equation.3. Touch the GRAPH button.4. Touch 2nd then CALC.5. Scroll to 3:minimum or 4:maximum and touch ENTER.6. Scroll to the left side of the graph where you see a max or min. Touch ENTER.7. Scroll to the right side of the graph where you see a max or min. Touch ENTER.8. Touch ENTER again.
Alg II Chapter 5 Notes Part 2.notebook
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Example 3: Maximize a Polynomial Model