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2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems. Chapter 2 Power, Polynomial, and Rational Functions. Warm-up. Describe end behavior using limits. Determine the zeros and state the multiplicity of any repeated zeros Find a few additional points, and .

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2-3 The Remainder and Factor Theorems

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  1. 2-3 The Remainder and Factor Theorems Chapter 2 Power, Polynomial, and Rational Functions

  2. Warm-up • Describe end behavior using limits. • Determine the zeros and state the multiplicity of any repeated zeros • Find a few additional points, and …graph the polynomial.

  3. Homework Check—Questions?

  4. Lesson 2-3: The Remainder and Factor theorems Objectives: • Dividing polynomials using long division • Zero remainder • Nonzero remainder • Division by degree 2 or higher • Dividing polynomials using synthetic division • Using the Remainder Theorem • Using the Factor Theorem

  5. Dividing Polynomials • Consider • If you know that this function has a zero at x = -6, you know that factored, • To divide polynomials to find that second-degree polynomial, we can use an algorithm similar to long division

  6. Divide by (x + 6)

  7. Example 2: Non-Zero Remainder • Divide by (2x – 1)

  8. Example 3: Divide by polynomial of Degree 2 or higher • Divide by

  9. Synthetic Division • A method of dividing a polynomial by a binomial factor in the form (x – c). • The polynomial must be written in standard form • Zeros must be used as placeholders for any mission powers. • For polynomials being divided by divisors that have degree 2 or higher, long division is used.

  10. Example 4: Divide using synthetic division

  11. Example 5: Divide using synthetic division

  12. The remainder Theorem • You can evaluate a polynomial function f(x) for x = c by using synthetic division dividing by (x – c). • This is called synthetic substitution. • Just like synthetic division, but only the last number in the last line is of interest.

  13. Example 6 : Real Estate • Research indicates that for each $10 decrease in rent of beachfront properties, 15 more units would be rented. • Weekly revenue …where x is # of $10 decreases • Use the remainder theorem to find the revenue from the properties if the property manager decreases the rent by $50.

  14. The Factor Theorem • Use the Factor Theorem to determine if binomials are factors of a given f(x) • If the remainder is a zero, then the divisor is a factor of the function. • If the remainder is not zero, then the divisor is not a factor of the function.

  15. Example 7: • Use synthetic division to determine if the binomials given are factors of f(x). Use the binomials that are factors to write a factored form of f(x).

  16. Example 8: • Use synthetic division to determine if the binomials given are factors of f(x). Use the binomials that are factors to write a factored form of f(x).

  17. Assignment: • p. 115, 1 - 13, odds, 19, 21, 27, 31 - 37 odds

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