1. Section 2.4 Dividing Polynomials; Remainder and Factor Theorems

2. Long Division of Polynomials and The Division Algorithm
Key Vocabulary:
Dividend
Divisor
Quotient
Remainder

4. Long Division of Polynomials
Solution is:

5. Long Division of Polynomials
Solution is:

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

7. Do Now Please
Divide using Long Division.

9. Example 5
Divide using Long Division.

10. Dividing Polynomials Using Synthetic Division
Vocabulary:
Synthetic Division

11. Synthetic Division To Divide a polynomial by x - c
Arrange the polynomial in descending powers, with a 0 coefficient for any missing term.
Write c for the divisor, x – c. To the right, write the coefficients of the dividend.
Write the leading coefficients of the dividend on the bottom row.
Multiply c times the value in the bottom row. Write the product in the next column in the second row.
Add the values in the new column, writing the sum in the bottom row.
Repeat this series of multiplications and additions until all columns are filled in.
Use the numbers in the LAST row to write the quotient, plus the remainder above the divisor. The degree of the fist term of the quotient is one less than the degree of the first term of the dividend. The final value in this row is the remainder.

12. ↓ Bring down 1 ↓ Add ↓ Add ↓ Add
Multiply 3 and 1
Quotient
Remainder
Multiply 3 and 7
Multiply 3 and 16

13. Comparison of Long Division and Synthetic Division of x3 + 4x2 - 5x + 5 divided by x - 3
List at least 3 things that you notice about the relationship between Long Division and Synthetic Division.

14. Steps of Synthetic Division dividing 5x3 + 6x + 8 by x + 2
Quotient
Remainder

15. Divide using synthetic division.
Example 7
Divide using synthetic division.
Quotient
Remainder

16. Using Synthetic Division instead of Long Division
Notice, that the divisor of all the Synthetic Division problems we have done have a degree of 1.
Thus:

17. The Remainder Theorem
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).
Remainder

18. Factor Theorem using the Remainder Theorem

19. They are the same!!!!!

20. Example 9
Use synthetic division and the remainder theorem to find the indicated function value.
They are the same!!!!!

21. Solve the equation 2x3 - 3x2 - 11x + 6 = 0 shows 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. So that PROVES -2 IS a zero! AND x + 2 IS a factor of the dividend!
Example 11
Solve the equation 5x2 + 9x – 2 = 0 given that -2 is a zero of f(x)= 5x2 + 9x - 2
So that PROVES -2 IS a zero!
AND x + 2 IS a factor of the dividend!

23. AND x - 5 IS a factor of the dividend!
Example 12
Solve the equation x3 - 5x2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x3 - 5x2 + 9x – 45.
So that PROVES 5 IS a zero!
AND x - 5 IS a factor of the dividend!

24. Review Time!!!!
Add these problems to your notes paper to help you review!
Additional Practice problems can be found on page 344 – 346 problems 47-81

25. (a)
(b)
(c)
(d)
(d)

26. (a)
(b)
(c)
(d)
(b)