Synthetic Division

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 of (linear) • If the divisor isn’t in the form x – c, then you must convert the expression to include subtraction. • X + 5 change to x – (-5) 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. Missing some terms

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 • 3x4 + – 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 3 0 -8 -11 1 Write all the coefficients of the expanded dividend here.

15. Step 3 2 3 0 -8 -11 1 3 “Drop” the 1st coefficient underneath the line.

16. Step 4 2 3 0 -8 -11 1 6 3 Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

17. Step 5 2 3 0 -8 -11 1 6 3 6 Add together the numbers in that column and write their sum underneath the line.

18. Step 6 2 3 0 -8 -11 1 6 12 3 6 Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

19. Step 7 2 3 0 -8 -11 1 6 12 8 -6 3 6 4 -3 -5 Repeat steps 5 and 6 until a number appears in the box underneath the last column.

20. Step 8 – Naming the Quotient 2 3 0 -8 -11 1 6 12 8 -6 3 6 4 -3 -5 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 3 6 4 -3 -5 The number in the box is the remainder. 3x3 + 6x2 + 4x – 3

22. Synthetic Division and the Factor Theorem • Conclusions:

23. Your Turn:

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

25. Your Turn: • On the Synthetic Division Practice handout, solve for the remaining roots for problems 1 – 4 and 10 – 12

26. 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

27. Reminder: Roots vs. Linear Factors Linear Factors Roots

28. 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