1. 5.2 The Factor Theorem & Intermediate Value Theorem

2. A polynomial P(x) has a factor x – c if and only if P(c) = 0.
Factor Theorem
A polynomial P(x) has a factor x – c if and only if P(c) = 0.

3. Ex 1) Use factor theorem to determine whether or not D(x) is a factor of P(x)
Plug in -1 and if you get = 0, it is a factor
Yes!
x + 1 is a factor

4. Ex 2) Use factor theorem to determine whether or not 3 is a zero of P(x)
If 3 is a zero, your remainder will = 0 3 ↓ 3 3 3 1 1 1 10
≠ 0
No!
3 is not a zero

5. Example 3 -1 1 -1 0 2 ↓ -1 2 -2 1 -2 2 x = -1, 1 + i, 1 - i
Better get 0! ↓ -1 2 -2 1 -2 2
x = -1, 1 + i, 1 - i
← Solve for x

6. Intermediate Value Theorem
If values of P(x) change from (+) to (–) or (–) to (+), there must be a 0 in between.
P(x) is (+)
Zero in between
(it crosses the x-axis!)
P(x) is (–)

7. Ex 4) Use I.V.T. & synthetic division to show that P(x) has a zero between -3 & -4 ↓ -3 6 9
(+) 1 -2 -3 15
Changes signs, so must be a 0 between -4 ↓ -4 12
-12 1 -3 3 -6
(–)

8. FYI – the Real Graph of
Zoomed in to see zero

9. Homework
#504 Pg – 49 odd