1. Aim #3. 3 How do we divide Polynomials
Aim #3.3 How do we divide Polynomials? What is the Remainder and Factor Theorems?

2. Long Division of Polynomials and The Division Algorithm

4. Long Division of Polynomials

5. Long Division of Polynomials

6. Long Division of Polynomials with Missing Terms
You need to leave a hole when you have missing terms. This technique will help you line up like terms. See the dividend above.

8. Example
Divide using Long Division.

9. Example
Divide using Long Division.

10. Dividing Polynomials Using Synthetic Division

12. Comparison of Long Division and Synthetic Division of X3 +4x2-5x+5 divided by x-3

13. Steps of Synthetic Division dividing 5x3+6x+8 by x+2
Put in a 0 for the missing term.

14. Using synthetic division instead of long division.
Notice that the divisor has to be a binomial of degree 1 with no coefficients.
Thus:

15. Example
Divide using synthetic division.

16. The Remainder Theorem

17. If you are given the function f(x)=x3- 4x2+5x+3 and you want to find f(2), then the remainder of this function when divided by x-2 will give you f(2)
f(2)=5

19. Example
Use synthetic division and the remainder theorem to find the indicated function value.

20. The Factor Theorem

21. Solve the equation 2x3-3x2-11x+6=0 given that 3 is a zero of f(x)=2x3-3x2-11x+6. The factor theorem tells us that x-3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor.
Another factor

22. Example
Solve the equation 5x2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x2 + 9x - 2

23. Example
Solve the equation x3- 5x2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x3- 5x2 + 9x – 45. Consider all complex number solutions.

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