5.2 The Factor Theorem & Intermediate Value Theorem

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Presentation transcript:

5.2 The Factor Theorem & Intermediate Value Theorem

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.

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

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 1 -2 -2 7 ↓ 3 3 3 1 1 1 10 ≠ 0 No! 3 is not a zero

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

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 (–)

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

FYI – the Real Graph of Zoomed in to see zero

Homework #504 Pg. 285 25 – 49 odd