pencil red pen highlighter notebook Work on example
pencil, red pen, highlighter, notebook Work on example #1 in today’s packet.
Recall from yesterday that we used long division when dividing polynomials. Example #1: Let P(x) = 2 x 3 + 3 x 2 – 15 x – 16 and d(x) = x – 3, find – . + – +
P(x) = 2 x 3 + 3 x 2 – 15 x – 16 d(x) = x – 3 There is a nice shortcut to long division called synthetic division Synthetic division is for divisors of the ________. x – k Let’s try the problem using this new method. form _____. coefficients of dividend _______ zeros of the divisor _______ *multiply* 2 bring down 3 3 – 15 – 16 +6 + 27 + 36 9 12 20 2 • • • coefficients of quotient q(x) ______ remainder r(x) _______
Example #2: Let P(x) = x 4 – 5 x 2 – 10 x – 12 and d(x) = x + 2, find . 0 for missing terms. Remember to put in ___ 1 – 2 1 • • Check : – 5 – 10 – 12 – 2 + 4 + 2 + 16 – 8 4 0 – 2 • • – 1
Example #3: Let P(x) = 2 x 3 + 5 x 2 – 7 x – 12 and d(x) = x + 3, find 2 – 3 2 • • 5 . – 7 – 12 – 6 + 3 + 12 – 1 – 4 0 0 (x + Since the remainder is ___, factor 3) is a _______ of P(x) = (______)(______) • zero of the graph of y = P(x). Likewise, (– 3, 0) is a _____ Check:
CONCLUSION : Factor Theorem P(x) has a factor (x – k) if and only if ________: 0 P(x) = 0 (i. e. , the remainder = ____). Remainder Theorem If P(x) is divided by (x – k), then the __________: 0 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) = 2 x 4 + 7 x 3 – 4 x 2 – 27 x – 18, find all the zeros of P(x). Step 1: divide by (x – 2). 2 7 +4 2 2 • • 11 – 27 – 18 + 22 + 36 + 18 – 4 18 9 • • P(x) = (______)(__________)
P(x) = (______)(__________) 2 x 3 + 11 x 2 + 18 x + 9 by (x + 3). Step 2: divide _________ 2 – 3 2 11 18 9 – 6 – 15 – 9 5 3 6 Try to factor quadratics • • • P(x) = (______)(______) 2 3 5 x +1 2 x 2 + 2 x P(x) = (______)(______)(_______) + 3 x +3 2 x
P(x) = (______)(______)(_______) 2, – 3, – 1, x = ________ Step 3: final all zeros of P(x). Step 4: Sketch a graph of y = P(x). y – 3 – 1 (0, – 18) 2 x
Work on the Synthetic Division & Finding Zeros Practice Worksheet
Day 10 Practice: Synthetic Division & Finding Zeros Practice 1. Find a) and 1 +2 – 12 – 5 8 +2 – 20 – 50 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) 1 +2 0 – 7 6 +2 +4 – 6 2 – 3 1 • • • -1 -3 3 2 P(x) = (______)(_____) P(x) = (______)(______) y To find the zeros, set P(x) = 0. 0 = (______)(______) – 3, 1, 2 x = _______ 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
- Slides: 14