1. Synthetic Division
1 March 2011

2. Synthetic Division
A trick for dividing polynomials
Helps us solve for the roots of polynomials
Only works when we divide by 1st degree (linear) polynomials
My degree can’t be larger than 1!

3. Synthetic Division

4. Your Turn
On the Synthetic Division – Guided Notes handout, complete problems 1 – 5. You will:
Decide if it’s possible to use synthetic division to divide the two polynomials

5. Division Vocab Review
Dividend
Divisor
Quotient

6. Preparing for Synthetic Division
Can only be used when the divisor is in the form
If the divisor isn’t in the form x – c, then you must convert the expression to include subtraction.
x – c

7. Preparing for Synthetic Division, cont.

8. Preparing for Synthetic Division, cont.
Polynomials need to be written in expanded, standard polynomial form.
Translation: If you’re missing terms, then you need to write them out as 0 times (*) the variable.

9. Preparing for Synthetic Division, cont.

10. Your Turn
On Synthetic Division - Guided Notes handout, write the dividend in expanded standard polynomial form for problems 6 – 10.
Write the divisor in the form x – c.

11. *Synthetic Division Steps
Example Problem:

12. Prep Step
Divisor x – c?
x – 2
Dividend in Expanded Standard Polynomial Form?
3x4 – 8x2 – 11x + 1
3x – 8x2 – 11x + 1
3x4 + 0x3 – 8x2 – 11x + 1

13. Step 1 2
Write the constant value of the divisor (c) here.

14. Step 2 2
Write all the coefficients of the expanded dividend here.

15. Step 3 2 3
“Drop” the 1st coefficient underneath the line.

16. Step 4 2 6 3
Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

17. Step 5 2 6
Add together the numbers in that column and write their sum underneath the line.

18. Step 6 2
Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

19. Step 7 2
Repeat steps 5 and 6 until a number appears in the box underneath the last column.

20. Step 8 – Naming the Quotient2
In the last row are the coefficients of the quotient in decreasing order. The quotient is one degree less than the dividend.

21. Step 8 – Naming the Quotient
The number in the box is the remainder.
3x3 + 6x2 + 4x – 3 Remainder -5

22. Your Turn
On the Synthetic Division – Guided Notes handout, solve for the quotient of problems 11 – 14 using synthetic division

23. Synthetic Division and the Factor Theorem
Conclusions:

24. Your Turn:
Using problems 1 – 12 on the Synthetic Division Practice handout (last night’s hmwk), identify which problems represent division by a factor/root and which problems represent division by NOT a factor root.

25. So What’s Next?
* To get the remaining roots, set the expression equal to 0, factor, and solve.

26. Your Turn:
On the Synthetic Division Practice handout, solve for the remaining roots for problems – 4 and 10 – 12

27. Rewriting the Original Polynomial
We can use the roots and linear factors to rewrite the polynomial
This form is called the product of linear factors
If you multiplied all the linear factors together, then you’d get the original polynomial

28. Reminder: Roots vs. Linear Factors

29. Product of Linear Factors
Product = Multiply
Product of linear factors = Multiply all the linear factors
Translation: Rewrite all the linear factors with parentheses around each factor
Helpful format for graphing polynomials
Product of Linear Factors