1. Division of Polynomials and the Remainder and Factor Theorems

2. 1. Divide Polynomials Using Long Division
2. Divide Polynomials Using Synthetic Division
3. Apply the Remainder and Factor Theorems

3. Divide Polynomials Using Long Division
Suppose that f(x) and d(x) are polynomials where
d(x) ≠ 0 and the degree of d(x) is less than or equal to the degree of f(x).
Then there exists unique polynomials q(x) and r(x)
such that
where the degree of r(x) is zero or of lesser degree than d(x).
Note: The polynomial f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.

4. Divide Polynomials Using Long Division
To begin the division: Always write the terms of the dividend and divisor in descending order with place holders for missing powers of x.

5. Example 1:
Use long division to divide.

6. Example 2:
Use long division to divide.

7. Example 3:
Use long division to divide.

11. 1. Divide Polynomials Using Long Division
2. Divide Polynomials Using Synthetic Division
3. Apply the Remainder and Factor Theorems

12. Divide Polynomials Using Synthetic Division
When dividing polynomials where the divisor is a binomial of the form (x – c) and c is a constant, we can use synthetic division.
Example:

13. Divide Polynomials Using Synthetic Division
The setup:
Step 1: Write the value of c in a box.
Step 2: Write the coefficients of the dividend to the right of the box.

14. Divide Polynomials Using Synthetic Division
Step 3: Skip a line and draw a horizontal line below the list of coefficients.
Step 4: Bring down the leading coefficient from the dividend and write it below the line.

15. Divide Polynomials Using Synthetic Division
The division:
Step 5: Multiply the value of c by the number below the line. Write the result in the next column above the line.

16. Divide Polynomials Using Synthetic Division
Step 6: Add the numbers in the column above the line, and write the result below the line.
Repeat steps 5 and 6 until all columns have been completed.

17. Divide Polynomials Using Synthetic Division
The right-most number below the line is the remainder. The other numbers below the line are the coefficients of the quotient in order by the degree of the term. Since the divisor is linear (1st-degree), then the degree of the quotient is 1 less than the degree of the dividend.

18. Example 4:
Divide using synthetic division.

19. Example 5:
Divide using synthetic division.

22. 1. Divide Polynomials Using Long Division
2. Divide Polynomials Using Synthetic Division
3. Apply the Remainder and Factor Theorems

23. Apply the Remainder and Factor Theorems
Remainder Theorem:
If a polynomial f (x) is divided by x – c, then the remainder is f (c).
Note: The value of f (c) is the same as the remainder we get from dividing f (x) by x – c.

24. Example 6:
Given , use the remainder theorem to evaluate

26. Example 7:
Use the remainder theorem to show that is a zero of .

27. Apply the Remainder and Factor Theorems
Polynomials may also be evaluated over the set of complex numbers rather than restricting x to the set of real numbers. A complex number a + bi is a zero of a polynomial f (x) if f (a + bi) = 0.

28. Example 8:
Use the remainder theorem to show that is a zero of .

30. Apply the Remainder and Factor Theorems
Let f(x) be a polynomial.
1. If f(c) = 0, then (x – c) is a factor of f(x).
2. If (x – c) is a factor of f(x), then f(c) = 0.

31. Example 9:
Use the factor theorem to determine if is a factor of

33. Example 10:
Factor given that
is a factor.

35. Example 11:
Write a polynomial of degree 3 that has the zeros 2
(of multiplicity 2) and –5.