1. January 10, 2012 At the end of today, you will be use long division and synthetic division to divide polynomials. Warm-up: The total revenue R earned (in dollars) from producing a gift box of candles is given by R(p) = -10p 2 + 800p where p is the price per unit (in dollars). a)Find the revenues when the prices per box are $20 and $30. b)Find the unit price that will yield a maximum revenue. What is the maximum revenue? Quiz 2.1 (on 2.1 and 2.2) Thursday! HW 2.3: Pg. 160 #57-62

2. Correct HW 5.7 1.C27a) x = ±5, b) odd mult; tp: 1 2.G29a) x = ±7; b) odd mult; tp: 1 3.H31a) x = -2, 1; b) odd; tp: 1 4.F33a) x = 0, 2 ±√3; odd, tp: 2 5.A35a) x = 0, 2; at 0 odd mult, at 2 even mult 6.E tp: 2 7.D37a) x = 0, ±√3; b) 0 odd, ±√3 even, tp: 4 8.B39a) no real zeros, tp: 1 41a) x = ±2, -3; odd mult; tp: 2

3. Lesson 2.3 Polynomial and Synthetic Division Polynomial and Synthetic Division is used to help find zeros of a polynomial function. For example: If the function f(x) 6x 3 – 19x 2 + 16x – 4 has one root x = 2, we know (x – 2) is one factor. To find the other factors, let’s use long division: 1. Divide first term by the first term of the divisor. 2. Multiply by the divisor, then subtract. 3. Bring down next term −

4. Continued… So, 4. List factors: - factor 5. FINALLY, solve for zeros Try one on your own: (6x 3 + 41x 2 – 9x – 14) ÷(2x + 1)

5. Using Synthetic Division *Useful if the coefficient of the divisor is 1 (5x 3 – 13x 2 + 10x – 8) ÷ (x – 2) Step 1: Write down the coefficients for each term in power descending order. 5 -13 10 -8 Step 2: Write the opposite constant of the divisor and bring down the first coefficient. 2 5 Step 3: Multiply the divisor and the number, then combine with the next. x 10 = -3 -6 4 8 0 So, (5x 3 – 13x 2 + 10x – 8) ÷ (x – 2) = 5x 2 – 3x + 4

6. Finding Roots using Synthetic Division cont… (x – 2)(5x 2 – 3x + 4) 4. List factors: Cannot factor, so use quadratic formula 5. FINALLY, solve for zeros x = 2 AND

7. Finding factors of a polynomial using repeated division Show that (x – 2) and (x + 3) are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x – 18 2 2 7 -4 -27 -18 2 4 11 22 18 36 9 18 0 2 11 18 9-3-3 2 -6 5 -15 3 -9 0 2x 3 + 11x 2 + 18x + 9 2x 2 + 5x + 3 Now Factor the last polynomial to get the other factors. (2x + 3)(x + 1) FINALLY: (x – 2)(x + 3)(2x + 3)(x + 1) Zeros: x = 2, -3, -3/2, -1