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Given this synthetic division problem, write the following: -3 1 -2 -9 18

w↑. Given this synthetic division problem, write the following: -3 1 -2 -9 18 -3 15 -18 1 -5 6 0 The polynomial d ivisor: The polynomial dividend: The polynomial quotient:. w↑ (review). Find the zeros/roots Find the x-intercepts

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Given this synthetic division problem, write the following: -3 1 -2 -9 18

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  1. w↑ • Given this synthetic division problem, write the following: -3 1 -2 -9 18 -3 15 -18 1 -5 6 0 • The polynomial divisor: • The polynomial dividend: • The polynomial quotient:

  2. w↑ (review) • Find the zeros/roots • Find the x-intercepts • List the factors • Describe the end behavior • Rough sketch of the graph • Check all info on graphing calculator

  3. Remainder and Factor Theorem Section 6.5

  4. Objectives • Identify the Remainder Theorem • Identify the Factor Theorem • Use them to completely factor and solve higher order polynomials

  5. Remainder Theorem • If , use synthetic division to divide by • Now, evaluate Notice anything?

  6. Remainder Theorem • Evaluate • Divide by Notice anything?

  7. Remainder Theorem • If a polynomial is divided by the remainder will be Putting it into your own words… • If you plugged “k” into the function, the value you would get after simplifying would be the remainder if you divided • You can plug a divisor’s zero into the function and see what the remainder would be if you divided by it

  8. Example • Evaluate • Divide by Here, the remainder theorem confirms that there would be no remainder for dividing by , so we know that would be a factor of the polynomial

  9. Factor Theorem • If a polynomial, , has a factor, , then • If you plugged “k” (which is the zero of the divisor that you know to be a factor of ), into the function – the resulting value after you simplify would be zero.

  10. Factor Completely • Factor given that • If what would the first factor of be? • Think about synthetic division

  11. Factor Theorem • Factor completely given that Steps to Follow • Use synthetic division with the given zero to find the other factors • Once you find the other factor, you can continue factoring if possible, and/or find the zeros of the function (if you’re told to)

  12. Finding zeros • and one zero is . Find the other zeros of the function algebraically. Steps to Follow • Use synthetic division with the zero that was given to find the factors • Then set equal to zero and solve for all zeros

  13. Example • If and one zero is , find the other zeros of the function

  14. Find all zeros • One zero is 4

  15. Find all zeros • One zero is 10

  16. Mixed Practice Factor the polynomial given that

  17. Mixed Practice Factor the polynomial given that

  18. Mixed Practice Given one zero, find the other zeros. • One zero is -3

  19. Mixed Practice Given one zero, find the other zeros. • One zero is -5 **Check Answer on Graphing Calculator

  20. Homework • Textbook page 357 • Numbers 41-51, 54

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