Sec 3.4 – Polynomial Functions Rational Root Theorem & Remainder Theorem Name:

Rene’ Descartes is commonly credited for devising the Rational Root Theorem.

The theorem states: Given a polynomial equation of the form

0 = 푎 푥 + 푎 푥 + 푎 푥 + … … … + 푎 푥 + 푎

Any rational root of the polynomial equation must be some integer factor of 푎

divided by some integer factor of 푎

Given the following polynomial equations, determine all of the “POTENTIAL” rational roots based on the Rational Root Theorem and then using a synthetic division to verify the most likely roots.

1. 푥 + 푥 − 8푥 − 12 = 0

2. 4푥 − 12푥 + 5푥 + 6 = 0

The Remainder Theorem suggests that if a polynomial function P(x) is divided by a linear factor (x – a) that the quotient will be a polynomial function, Q(x), with a possible constant remainder, r , which could be written out as:

푷(풙) = (풙 − 풂) ∙ 푸(풙) + 풓 If this seems a little complicated consider a similar statement but just using integers. For example (using the same colors to represent similar parts), ퟕퟎ ÷ ퟔ = ퟏퟏ퐰퐢퐭퐡퐫퐞퐦퐚퐢퐧퐝퐞퐫ퟒ which could also be rewritten as: ퟕퟎ = ퟔ ∙ ퟏퟏ + ퟒ

The Remainder Theorem also leads to another important idea, The Factor Theorem. To state the Factor Theorem, we only need to evaluate P(a) from the Remainder Theorem.

푷(풂) = (풂 − 풂) ∙ 푸(풂) + 풓 :Substitute “a” in for each “x”

푷(풂) = (ퟎ) ∙ 푸(풂) + 풓 :Simplify (a – a) = 0

푷(풂) = 풓 :Simplify 0∙Q(a) = 0 This is an important fact that basically states the remainder of the statement 푷(풙) ÷ (풙 − 풂)is 푷(풂).

Potential Rational Roots: Potential Rational Roots:

M. Winking Unit 3-4 page 50

Using the Remainder or Factor Theorem answer the following.

3. Using Synthetic Division evaluate 푥 + 푥 − 8푥 − 12 when x = 3

4. Use Synthetic Division to find the remainder of (푥 + 푥 − 8푥 − 12) ÷ (푥 − 3)

5. Using Synthetic Division evaluate 푓(−2) given 푓(푥) = 3푥 + 7푥 − 8푥 + 12.

6. Use Synthetic Division to determine the quotient of 푓(푥) and 푔(푥), given 푓(푥) = 3푥 + 7푥 − 8푥 + 12 and 푔(푥) = 푥 + 2

7. Given 풇(풙) = (풙 + ퟓ) ∙ 푸(풙) + ퟖ, evaluate 풇(−ퟓ).

8. Given 풇(풙)(풙 ퟔ) = 푸(풙)퐰퐢퐭퐡퐚퐫퐞퐦퐚퐢퐧퐝퐞퐫ퟑ,

evaluate 푓(6).

9. Consider 푓(푥) = 2푥 − 1푥 + 3푥 + 4 and

that 푓(푏) = 5

What value should be in box labeled “a”?

10. Consider 푔(푥) = 푥 + 3푥 − 2푥 + 4 and Justin used synthetic division to divide(푥 + 3푥 − 2푥 + 4) ÷ (푥 − 푘). His work is partially shown below. Using this information determine푓(푘).

M. Winking Unit 3-4 page 51a

Using any available techniques determine the following (find exact answers).

11. Find all of the solutions to the polynomial equation 푥 − 3푥 + 6푥 − 12푥 + 8 = 0

12. Find all zeros of the polynomial function 푓(푥) = 푥 − 4푥 + 푥 + 8푥 − 6

M. Winking Unit 3-4 page 51b