6.1 Polynomial Functions

Polynomials

A polynomial is a sum of terms whose exponents are whole numbers (not fractions or negative numbers).

Polynomials:

y = x3 + 4x2 – 2x + 1

y = x

y = 10

257

2

3xxxy

Not Polynomials:

12 xxy

61

31

15 xxy

xxy

315

Classifying Polynomials

A polynomial is said to be in standard form when the terms are in descending order by degree.

y = x3 + 4x2 – 2x + 1

257

2

3xxxy

What is the degree of the polynomial? What is the leading coefficient?

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 ModelsUsing 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 ModelsThe 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 65

1985 70

1990 60

1995 55

2000 64

Multiplying PolynomialsMultiply:

2312 xx This is just FOIL

2346 2 xxx

26 2 xx

961 2 xxx This is just like FOIL

96

962

23

xx

xxx9157 23 xxx

Multiplying PolynomialsMultiply:

2312 22 xxxx

234 23 xxx 2795 234 xxxx

45232 23 xxxx

xxxx 81042 234

xxx 462 23

232 xx

121563 23 xxx127472 234 xxxx

Multiplying PolynomialsMultiply:

2435 22 xxxx

234 10205 xxx 61417215 234 xxxx

27354 23 xxxx

xxxx 828124 234

xxx 24 23

6123 2 xx

1035155 23 xxx104343174 234 xxxx

Multiplying PolynomialsMultiply:

41x 22 11 xx

1212 22 xxxx

234 2 xxx

xxx 242 23

122 xx

1464 234 xxxx