Polynomials Polynomials Types of Polynomials About Polynomials Graphing

Polynomials

Polynomials • • Types of Polynomials About Polynomials Graphing Polynomials Evaluating Polynomials

Types of Polynomials • • Monomial Binomial Trinomial Polynomial

Types of Polynomials A polynomial is a broad term for a function that only has many terms with positive integer exponents. is not a polynomial.

About Polynomials • Standard form of a polynomial is when the terms of a polynomial are written in descending order of exponents. is not in standard form. is in standard form.

Rewrite the Polynomials in Standard Form

About Polynomials • The leading coefficient is number of the term with the highest degree.

Identify the Leading Coefficient

About Polynomials • The degree of the polynomial is the greatest exponent.

Identify the Degree of the Polynomial

Graphs of Polynomials y 10 8 6 4 2 x -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10

Even Exponents • Positive Leading Coefficient 10 -8 -6 -4 • Negative Leading Coefficient y 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 x -10 -8 -6 -4 -2 y 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 4 6 8 10 x

Odd Exponents • Positive Leading Coefficient 10 -8 -6 -4 • Negative Leading Coefficient y 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 x -10 -8 -6 -4 -2 y 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 4 6 8 10 x

Graph on your Calculator

Evaluating Polynomials Using Synthetic Substitution

Why not just “plug it in? ” • Don’t really need a calculator. • We’ll use the idea to make dividing polynomials, which can be difficult, an easier task.

1. Rewrite the coefficients and put the “ 2” we’re evaluating to the side. 2 -3 -4 8 2|

2. Bring down the “ 2”. 2 -3 2| 2 -4 8

3. Multiply the “ 2” on the side by the “ 2” that was brought down and put product below “-3”. 2 -3 4 2| x 2 = -4 8

4. Add -3 and 4 and write sum below. 2 2| 2 -3 +4 1 -4 8

5. Multiply the “ 2” on the side by the “ 1” that was brought down and put product below “-4”. 2 2| 2 x -3 +4 1 -4 2 = 8

6. Add -4 and 2 and write sum below. 2 2| 2 -3 4 1 -4 +2 -2 8

7. Multiply the “ 2” on the side by the “-2” that was brought down and put product below “ 8”. 2 2| 2 x -3 4 1 -4 +2 -2 8 -4 =

8. Add 8 and -4 and write sum below. 2 -3 -4 8 2| 4 +2 -4 2 1 -2 4

You try one!

Try a Tricky One!