1. M3U4D3 Warm Up Without a calculator, divide the following Solution: 49251 NEW SEATS

2. Homework Check:

5. (3a+8b)(3a-8b)

6. M3U3D3 Synthetic Division OBJ: To solve polynomial equations involving division.

7. Synthetic Division - To use synthetic division: There must be a coefficient for every possible power of the variable. The divisor must have a leading coefficient of 1. divide a polynomial by a polynomial 1 SWC to Demonstrate using long division first!

8. Step #1: Write the terms of the polynomial so the degrees are in descending order. Since the numerator does not contain all the powers of x, you must include a 0 for the

9. Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. Since the divisor is x-3, r=3 50-416

10. 5 Step #3: Bring down the first coefficient, 5.

11. 5 Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add. 15

12. 5 Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 45 41

13. 5 15 45 41 Step #5 cont.: Repeat the same procedure. 123 124 372 378 Where did 123 and 372 come from?

14. Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend. 5 15 45 41 123 124 372 378

15. The quotient is: Remember to place the remainder over the divisor!

16. Ex 2: Step#1: Powers are all accounted for and in descending order. Step#2: Identify r in the divisor. Since the divisor is x+4, r=-4. SWC to Demonstrate using long division first!

17. Step#3: Bring down the 1st coefficient. Step#4: Multiply and add. -5 Step#5: Repeat. 20 4-4 0 8 10-210

18. Try this one:

19. SYNTHETIC DIVISION: Practice [1] [2][3] -4 1 2 -5 12 -4 8 -12 1 -2 3 0 -1 1 -5 -13 0 10 -1 6 7 -7 1 -6 -7 7 3

20. The Remainder Theorem The remainder obtained in the synthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. That is, to evaluate a polynomial function f (x) when x = k, divide f (x) by x – k. The remainder will be f (k).

21. Example Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x = –2. f (x) = 3x 3 + 8x 2 + 5x – 7 Solution: Using synthetic division, you obtain the following.

22. Example – Solution Because the remainder is r = –9, you can conclude that f (–2) = –9. This means that (–2, –9) is a point on the graph of f. You can check this by substituting x = –2 in the original function. Check: f (–2) = 3(–2) 3 + 8(–2) 2 + 5(–2) – 7 = 3(–8) + 8(4) – 10 – 7 = –9 r = f (k) cont’d

23. REMAINDER THEOREM: Given a polynomial function f(x): then f(a) equals the remainder of Example: Find the given value 2 1 3 - 4 -7 2 10 12 1 5 65 Method #1: Synthetic Division Method #2: Substitution/ Evaluate [A] [B] -3 1 0- 5 8 -3 -39 -12 12 1 -3 4 -4 9

24. Classwork 1. Algebra 2 Notes: Synthetic Division Document Camera

25. Classwork 2. M3U3D3 Using the Remainder Theorem to Prove Zeroes Document Camera

26. Homework U3D3 Synthetic Division and the Remainder Theorem.