Name:
Date:
Period:
Operations with Polynomials and
Curve Fitting with Polynomials
Lessons: 6-2, 6-3, 6-9
Packet 8
Tennessee State Standards
SPI 3101.3.1 Add, subtract and multiply
polynomials; divide a polynomial by a lower
degree polynomial.
SPI 3103.5.3 Analyze patterns in a scatter-plot and
describe relationships in both linear and non-linear
data.
Common Core State Standards
F-BF-1b. Combine standard function types using
arithmetic operations. For example, build a
function that models the temperature of a cooling
body by adding a constant function to a decaying
exponential and relate these functions to the
model.
S-ID-6a Fit a function to the data; use functions
fitted to the data to solve problems in the context
of the data.
Name:
Date:
Lesson 6-2
Multiplying Polynomials
Period:
p.1
Warm-Up: Multiply Coefficients and Add Exponents of Like Bases.
x(x3)
2(5x3)
xy(7x2)
3x2(-x5)
Monomial times a Polynomial
4y2(y2 + 3)
fg(f4 + 2f3g – 3f2g2 + fg3)
Binomial times a Binomial (FOIL)
(x + 3) (x – 5)
(x – 4) (x + 4)
Polynomial times a Polynomial
(x – 3)(2 – 5x + x2)
(y2 – 7y + 5)(y2 – y – 3)
Name:
Lesson 6-2
Date:
Multiplying Polynomials
Period:
p.2
Binomial Expansion
( x + y)2
(a + 2b)3
Part B: Homework
1. 7x(2x + 3)
2. (x + 1)4
3. (x – 1)(2x2 + 3)
4. (x – 6)(x4 – 2x3 + 1)
5. 2xy(3y2 – xy + 7)
6. (x4 + x2y)(x3 + y2)
Name:
Date:
Lesson 6-3
Dividing Polynomials
Period:
p.3
Review of Long Division
161 ÷ 7 =
128 ÷ 9 =
Using Long Division to Divide Polynomials
(4𝑥 2 + 3𝑥 3 + 10) ÷ (𝑥 − 2)
Write the dividend in standard form, including missing terms!
Write division in the same way you would when dividing numbers.
Name:
Lesson 6-3
Date:
Dividing Polynomials
Period:
p.4
Part A: Guided Practice
(-y2 + 2y3 + 25) ÷ (y – 3)
Synthetic Division- Another Method
(2x2 + 7x + 9) ÷ (x + 2)
(15x2 + 8x – 12) ÷ (3x + 1)
Name:
Lesson 6-3
Date:
Synthetic Division
Period:
p.5
Part A: Guided Practice
(x4 – 2x3 + 3x + 1) ÷ (x – 3)
1
(3x2 + 9x – 2) ÷ ( x − )
3
Name:
Lesson 6-3
Date:
Dividing Polynomials
Period:
p.6
Part B: Homework
Use LONG division.
1. (y4 + 9y2 + 20) ÷ (y2 + 4)
Use SYNTHETIC division.
2. (6x3 – 14x2 + 10x – 4) ÷ (x – 1)
Use LONG division.
3. (60 – 16y2 + y4) ÷ (10 – y2)
Use SYNTHETIC division.
4. (x4 + 6x3 + 6x2) ÷ (x + 5)
Name:
Date:
Adding/Subtracting Polynomials
Period:
p. 7
When adding polynomials, simply ______________ __________ _____________.
1. (x3 – 2x2 + 4x – 2) + (x2 + 3x + 5)
2. (3x4 + x2 – 5x + 8) + (-5x4 + 3x3 + 7x2 – 2x – 1)
When subtracting polynomials, _________________ the subtraction sign and
then __________________ _____________ _____________.
1. (3x4 + 5x2 – 4x + 1) – (2x4 – x3 + 2x2 + 3x – 2)
2. (-2x3 + 8x2 + x – 3) – (5x3 + 6x2 – 4x + 3)
Name:
Date:
Adding/Subtracting Polynomials
Part B: Homework
1. (2x2 + 6x + 5) + (3x2 - 2x – 1)
2. (x4 + 2x3 – 4x + 6) + (-3x4 + 4x3 +1)
3. (4x3 + 2x2 – 4x + 1) – (x3 – 4x2 + 5x + 5)
4. (4x3 – 4x2 + 3x – 1) + (2x2 + x – 3)
5. (-3x2 + 2x – 4) – (6x2 – 3x – 3)
Period:
p. 8
Name:
Date:
Lesson 6-9
Period:
Curve Fitting
p. 9
To create a mathematical model for data, you will need to figure out what type of
function is ______________________________. Finite ________________ can be
used to identify the __________________ of any polynomial data.
This chart will help you decide which type of function to use:
Function Type
Linear
Quadratic
Cubic
Quartic
Quintic
Constant Finite Differences
Degree
Example:
x
y
2
-2
5
0
8
2
11
4
14
6
17
8
1. Find the differences in the y-values.
x
y
-6
-30
-4
15
-2
30
0
34
2
41
4
60
Name:
Date:
Lesson 6-9
Period:
Curve Fitting
p.10
2. Find the differences in the y-values.
X
y
-2
-10
-1
-4
0
-1.4
3. Find the differences in the y-values.
x
4
6
8
Y
-2
4.3
8.3
1
0
10
10.5
2
2.4
12
11.4
3
8
14
11.5
Once you know which type of function to use, use the calculator to write a
function.
1. Enter data on L1 and L2 (Stat Edit)
2. Go to Stat,Calc, and choose the appropriate type.
3. Enter- now put the function’s equation together.
**It can be harder to use the finite differences in real-life problems. If no finite
difference exists, use R2 to compare different models. Choose the model with
the highest value of R2. Make sure you turn the diagnostics on!
1. The table shows the population of a city from 1960 to 2000. Write a
polynomial function for the data. (Use 0 for 1960)
Year
1960
1970
1980
1990
2000
Population 4,267
5,185
6,166
7,830
10,812
(thousands)
Name:
Date:
Lesson 6-9
Curve Fitting
Period:
p.11
2. The table below shows the opening value of a stock on the first day of
trading in various years. Use a polynomial model to estimate the value of
the first day of trading in 2000. (Use 0 for 1994)
Year
1994
1995
1996
1997
1998
1999
Price ($) 683
652
948
1306
863
901
Type of Model: ____________________________________
Equation: _________________________________________
Estimate the value of the first day of trading in 2000: ______
3. The table below shows the number of infected patients at various stages of
a flu outbreak. Use a polynomial model to estimate the number of infected
patients after 120 hours.
Time (h) 12
24
48
96
144
240
Patients 21
301
679
973
562
320
Type of Model: _______________________________
Equation: ____________________________________
Estimate the number of infected patients after 120 hours: _________
Name:
Date:
Lesson 6-9
Curve Fitting
Period:
p.12
Part B: Homework
1. Use finite differences to determine the
degree of the polynomial that best
describes the data.
x 8
10 12
14
16
18
y 7.2 1.2 -8.3 -19.1 -29 -35.8
2. Write a polynomial function
using the given data.
x
y
5
30
10
34
15
36
20
36
25
34
Type of model:
Equation:
3. The table shows the population of a
bacteria colony over time. Write a
polynomial function for the data and
use it to estimate the number of
bacteria after 7 hours.
Time (h) 1 2
3
4
5
Bacteria 44 112 252 515 949
Type of model:
Equation:
Estimate the number of bacteria after 7
hours.
4. Use finite differences to
determine the degree of the
polynomial that best describes
the data.
x
y
-2 -3
-1 1
0
4.3
1
6.9
2
8.8
3
10
Name:
Date:
Lesson 6-2, 6-3, 6-9
Homework continued
Period:
p.13
5. Divide (5a2b- 3ab2 – 2b3) by
(ab – b2).
6. Use synthetic substitution to
evaluate the polynomial for
the given value. Show your
work.
P(x) = 4x3 – 5x2 + 3 for x = -1
7. Use synthetic substitution to
evaluate the polynomial for
the given value.
4
P(x) = 25x2 – 16 for x =
8. Is (x + 3) a factor of
3x3 + 5x2 +2x – 12? Show your
work.
5