Work on example #1 in today’s packet.

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Presentation transcript:

Work on example #1 in today’s packet. pencil, red pen, highlighter, notebook M3D15 Have out: Bellwork: Work on example #1 in today’s packet.

Introduction to Synthetic Division Recall from yesterday that we used long division when dividing polynomials. Example #1: Let P(x) = 2x3 + 3x2 – 15x – 16 and d(x) = x – 3, find . – + – + – +

coefficients of quotient q(x) P(x) = 2x3 + 3x2 – 15x – 16 d(x) = x – 3 There is a nice shortcut to long division called ________________. Synthetic division is for divisors of the form _____. Let’s try the problem using this new method. synthetic division x – k coefficients of dividend 2 3 –15 –16 _____________ bring down + 6 + 27 + 36 zeros of the divisor 3 2 9 12 _____________ 20 *multiply* • • • remainder r(x) _____________ coefficients of quotient q(x) ____________

Example #2: Let P(x) = x4 – 5x2 – 10x – 12 and d(x) = x + 2, find . Remember to put in ___ for missing terms. 1 –5 –10 –12 –2 + 4 + 2 + 16 –2 1 –2 –1 –8 4 • • • • Check:

 Example #3: Let P(x) = 2x3 + 5x2 – 7x – 12 and d(x) = x + 3, find . Since the remainder is ___, (x + 3) is a _______ of P(x). 2 5 –7 –12 factor –6 + 3 + 12 –3 2 –1 –4  P(x) = (______)(___________) • • • Likewise, (–3, 0) is a _____ of the graph of y = P(x). zero Check:

CONCLUSION: Factor Theorem _______________: P(x) has a factor (x – k) if and only if P(x) = 0 (i.e., the remainder = ____). Remainder Theorem ___________________: If P(x) is divided by (x – k), then the remainder = ____.

Using Synthetic Division to Find the Zeros of P(x) Example #4: Given that (x – 2) and (x + 3) are factors of P(x) = 2x4 + 7x3 – 4x2 – 27x – 18, find all the zeros of P(x). Step 1: divide by (x – 2). 2 7 –4 –27 –18 + 4 + 22 18 + 36 + 2 2 11 18 9  • • • • P(x) = (______)(____________________)

Try to factor quadratics P(x) = (______)(____________________) Step 2: divide __________________ by (x + 3). 2x3 + 11x2 + 18x + 9 2 11 18 9 6 5 Try to factor quadratics –6 –15 –9 3 2 –3 2 5 3  • • • x + 1 P(x) = (______)(______)(____________) 2x 2x2 + 2x + 3 + 3x + 3 P(x) = (______)(______)(______)(_______)

P(x) = (______)(______)(______)(_______) Step 3: final all zeros of P(x). 2, –3, –1, x = ________________ Step 4: Sketch a graph of y = P(x). y x –3 –1 2 (0, –18)

Work on the Synthetic Division & Finding Zeros Practice Worksheet

Day 10 Practice: Synthetic Division & Finding Zeros Practice a) and 1 –12 –5 8 +2 –20 –50 +2 1 –10 –25 –42 • • •

 2. Given each P(x) and factors of P(x), find all the zeros of P(x). a) , and factor (x – 2) -3 2 1 –7 6 -1 3 +2 +4 –6 P(x) = (______)(__________) +2 1 2 –3  P(x) = (______)(______)(______) • • • x y To find the zeros, set P(x) = 0. 0 = (______)(______)(______) –3, 1, 2 x = _____________

Check your answers: 1. a) b) c) d) 2. a) x = –3, 1, 2 b) x = –4, –2, 6 c) x = –2, 1, 5

Finish the worksheets