6.1 - Polynomials

Different types of modeling

Linear Model Quadratic Model Cubic Model

What is a polynomial?A polynomial is the sum or difference of monomials.

DEGREE NAME EXAMPLE # terms NAME (term)

0 Constant 6 1 Monomial

1 Linear x+1 2 Binomial

2 Quadratic 3x² + x – 4 3 Trinomial

3 Cubic 4x³ + x² 2 Binomial

POLYNOMIAL

Y = 2x³ - 5x² - 2x + 5

CUBICTERM

QUADRATICTERM

LINEARTERM

CONSTANT

LEADINGCOEFFICIENT

Match each polynomial with a description of it’s graph

Linear, binomial

3)( xxf

Quadratic, trinomial

Cubic, polynomial

3)( 2 xxxf

127)( 23 xxxxf

Quintic, monomial

Quartic, trinomial

5)( xxf

33)( 4 xxxf

Example

)3x5x2x5()1x9x8x7( 2535

3x5x2x51x9x8x7 2535

2x4x2x8x2 235

Write the polynomial from greatest degree to smallest degree

Classification: This is a Polynomial of 5 terms

Let’s try one

)9314()23512( 343 xxxx

54534 34 xxx

Simplify. Classify the result by the number of terms

Polynomial of 4 terms

POLYNOMIALS - MultiplyingEX = FOIL (2 Binomials)

(3x + 1)(2x – 5)

6x² 5-

First

Outer

Inner

Last

- 15x 2x+

First

Outer

Inner

Last

POLYNOMIALS - MultiplyingEX = BOX (2 Binomials)

(3x + 1)(3x – 7) 3x 1

3x 9x² 3x

-21x -7

= 9x² + 3x - 21x - 7

= 9x² -18x - 7

-7

Let’s try one

1)52( 3 x

Let’s try one

1)52( 3 x

POLYNOMIALS - MultiplyingEX = FOIL (Polynomials)

(x² - 3x + 1)(x² + x – 3)

x4

9x+- 3x³ 3x²-+ x³ - 3x²

3-+ x² x+

= x4 – 2x³ - 5x² + 10x - 3

POLYNOMIALS - MultiplyingEX = BOX (Polynomials)

(x² - 3x + 1)(x² + 5x – 4)

x² -3x 1

x² 5x -4

x4 +2x³

-18x² +17x - 4x4 -3x³ x²

5x³ -15x² 5x

-4x² 12x -4