Unit 3 Lesson 11: The Factor and Remainder Theorems
Objective: ___________________________________________________________________
1.
Do Now: Complete the algebraic example box using your preferred method: the area
model or synthetic division.
2.
Group Practice: Answer the following communication questions within your groups
using the graph or function provided.
Algebraic Example
Directions: If 𝑓(𝑥) = 𝑥 3 − 12𝑥 − 16, find 𝑓(𝑥) ÷ (𝑥 + 2) using synthetic division or an area model.
Hint: Check for missing terms.
𝑓(𝑥) ÷ (𝑥 + 2) = ______________________________
Graphical
Communication
The following graph represents the function
1.
Directions: Use 𝒇(𝒙) = 𝒙𝟑 − 𝟏𝟐𝒙 − 𝟏𝟔 to answer the
𝒇(𝒙) = 𝒙𝟑 − 𝟏𝟐𝒙 − 𝟏𝟔 from the Do Now. question questions below.
What are the x-intercept(s) of f(x)? (___, 0) (___, 0)
2.
What is the remainder when you divide f(x) by (x + 2)?
Remainder = ______
3.
What is the remainder when you divide f(x) by (x + 1)?
Remainder = ______
What is the remainder when you divide f(x) by (x – 4)?
Remainder = _______
4.
R
What is the remainder when you divide f(x) by (x – 1)?
Remainder = ______
5.
Find the following:
f(-2) = ___
6.
area m
f(-1) = ___ f(4) = ___ f(1) = ___
Predict what is the remainder when you divide f(x) by
(x – 2) WITHOUT doing synthetic division or using an
model? Justify your prediction.
The _____________ when you divide _____________
_____________ by ____________ is _____________.
I made my prediction by ________________________.
1.
Unit 3 Lesson 11: The Factor and Remainder Theorems
𝒈(𝒙) = 𝒙𝟒 − 𝟏𝟎𝒙𝟐 + 𝟗
Graphical
Communication
The following graph represents the function1.g(x)
from above.
2.
What are the x-intercept(s) of g(x)?
(___, 0) ( ___, 0) (___, 0) (___, 0)
2.
What is the remainder when you divide g(x) by (x - 3)?
Remainder = _______
3.
What is the remainder when you divide g(x) by (x + 2)?
Remainder = ______
5.
Find the following using the graph to the left:
g(3) = ________
6.
m
g(-2) = _________
Predict what is the remainder when you divide g(x) by
(x + 4) without doing synthetic division or using an area
model? Justify your prediction.
The _____________ when you divide _____________
_____________ by ____________ is _____________.
I made my prediction by ________________________
_____________________________________________.
Summary
Use the two examples above to come up with the definition of the remainder theorem and factor theorem.
Use the word bank to complete the sentence frames below.
Word bank: factor, remainder, plugging in, value, x-intercept, and zero.
Explain in your own words what is the remainder theorem?
__
You can find the ______________________ without using synthetic division or the area model when
ng two polynomial functions by ___________________________________________________________________.
2. Explain in your own words what is the factor theorem?
__
One polynomial function is a ______________________ of another polynomial function when the
____________________ is _____________________. If a value is a factor then it is an ______________________
on the graph of the function.
Unit 3 Lesson 11: The Factor and Remainder Theorems
Unit 3 Lesson 11 Homework
Directions: Divide f(x) by g(x) using synthetic division or an area model, then evaluate
f(x) at each of the following values.
f (x) = 3x 2 + 2x - 5
1a.
2a.
g(x) = x - 4
g(x) = x -1
1b. Find f(1) = __________.
Is g(x) a factor of f(x)? Circle one: YES
3a.
f ( x)  2 x 3  13 x 2  8 x  42
2b. Find f(4) = __________.
NO
Is g(x) a factor of f(x)? Circle one: YES
NO
4a.
f ( x)  2 x 3  9 x 2  8
f (x) = -5x 4 - 9x 3 + 7x 2 + 9x - 2
g ( x)  x  1
g(x) = x -1
4b. Find f(1) = __________.
3b. Find f(-1) = __________.
Is g(x) a factor of f(x)? Circle one: YES
NO
Is g(x) a factor of f(x)? Circle one: YES
NO