1. 6.3 Synthetic Division

2. Here is a quick way to evaluate the value of a function for a particular value of x. It is called synthetic substitution.
Evaluate for x = 3
129 15 39 5 13 43
124
The value of the function is 124 when x = 3

3. Use synthetic substitution to find the value of y when x = 2
The value of the function is -100 when x = 2

4. Use synthetic substitution to find the value of y when x = -2
The value of the function is 0 when x = -2
This means that x = -2 is a root of the function.

5. Since x = -2 is a root of the function, we know that….
Divide x4 + 5x3 –3x2 +45x –54 by x + 2 to determine what the something in the parentheses is:

6. Divide

7. Compare
Now compare this result to what you got when you did synthetic substitution. What do you notice?
Synthetic Substitution is the same as Long Division!!!!
(Except that it only works for linear factors)

8. Conclusion:
Since synthetic substitution, synthetic division, and long division are essentially the same process….
We can use synthetic division in most cases instead of long division.

9. Divide:

10. Divide:
Since the remainder is zero (x+3) is a factor.

11. Divide:
Since the remainder is zero (x - 9) is a factor.

12. What would happen if your divided repeatedly?
This polynomial has roots at x = 1 and –2 (and no others)

13. What would happen if your divided repeatedly?
This polynomial has roots at x = 1 and –2 (and no others)
Extra Space for division

14. Factor completely using synthetic division.
This polynomial has roots at x = 2 and 3 (and no others)

15. Factor completely using synthetic division.
This polynomial has roots at x = 2 and 3 (and no others)
Extra Space for division