Transcript 6.1 Polynomial Operations

6.1 Polynomial
Functions
Polynomials
A polynomial is a sum of terms whose exponents are whole
numbers (not fractions or negative numbers).
Polynomials:
Not Polynomials:
y = x3 + 4x2 – 2x + 1
y  x  x 1
y=x
y = 10
3 7
y  x  x 5  x 2
2
2
y  15 x
1
3
x
3
y  15 x 
x
1
6
Classifying Polynomials
A polynomial is said to be in standard form when the terms are
in descending order by degree.
What is the degree of the polynomial? What is the
leading coefficient?
y = x3 + 4x2 – 2x + 1
3 7
y  x  x 5  x 2
2
Adding Polynomials
To add polynomials, just combine like terms:
(8x3 – 3x2 – 2x + 9) + (2x3 + 6x2 – x + 1) =
10x3 + 3x2 – 3x + 10
(12x4 – 5x2 + x + 7) + (2x3 + 6x2 – x + 2) =
12x4 + 2x3 + x2 + 9
Subtracting Polynomials
To subtract polynomials, combine like terms.
(Just be careful with the signs.)
(8x3 – 3x2 – 2x + 9) - (2x3 + 6x2 – x + 1) =
6x3 - 9x2 – x + 8
(12x4 – 5x2 + x + 7) - (2x3 + 6x2 – x + 2) =
12x4 - 2x3 - 11x2 + 2x + 5
Comparing Models
Using a graphing calculator, determine whether a
linear model, a quadratic model, or a cubic model
best fits the values in the table.
X
0
5
10
15
20
Y
10.1
2.8
8.1
16.0
17.8
X
0
2
4
6
8
Y
2.8
5
6
5.5
4
Comparing Models
The table shows data on the number of employees
that a small company had from 1975 to 2000. Find
a cubic function to model the data. Use it to
estimate the number of employees in 1998.
Year
Number of
Employees
1975
60
1980
1985
65
70
1990
60
1995
2000
55
64
Multiplying Polynomials
Multiply:
2x  13x  2 
This is just FOIL
6 x  4 x  3x  2 
2
6x 2  x  2 
x  1x 2  6x  9 
x 3  6x 2  9x
x  6x  9
2
This is just like FOIL
 x  7 x  15x  9
3
2
Multiplying Polynomials
Multiply:
x
2


 2 x  1 x 2  3x  2 
x 4  3x 3  2 x 2
2x  6x  4x 
3
2
x  5x  9 x  7 x  2
4
3
2
x 2  3x  2
2x  3x 3  2x 2  5x  4 
2 x 4  4 x 3  10x 2  8x
3x 3  6 x 2  15x  12
 2 x 4  7 x 3  4 x 2  7 x  12
Multiplying Polynomials
Multiply:
5x
2


 x  3 x 2  4x  2 
5x 4  20x 3  10x 2
x  4 x  2 x  5x  21x  17x  14x  6
3
2
4
3
2
3x 2  12x  6
4x  5 x 3  3x 2  7 x  2 
 4 x 4  12x 3  28x 2  8x
5x 3  15x 2  35x  10
 4 x 4  17x 3  43x 2  43x  10
Multiplying Polynomials
Multiply:
x
x  14  x  12 x  12 
2


 2x  1 x 2  2x  1 
x 4  2x 3  x 2
2x 3  4x 2  2x
x 2  2x  1
 x 4  4x 3  6x 2  4x  1