4.3 Introduction to Polynomials 4/24/2017 POLYNOMIALS Introduction to Polynomials 4.3 S KM & PP Kathy Monaghan & Pat Peterson - AIM

The word “Polynomial” means “many names” or “many terms”. What is a Polynomial? The word “Polynomial” means “many names” or “many terms”. A “term” is a “monomial” and has the following form: a is a Real Number (constant) n is a non-negative integer. 4.3 S KM & PP

A Polynomial is... A Polynomial is one or more monomials (terms) combined by addition or subtraction. Here’s an example: 4.3 S KM & PP

Standard Form for a Polynomial is... Usually, polynomials are written in STANDARD FORM where the terms are listed so that the powers of the variable are decending (largest to smallest.) 4.3 S KM & PP

Polynomial Classification by the Number of Terms. A POLYNOMIAL is composed of the sum or difference of TERMS. A MONOMIAL is a single term. A BINOMIAL has two terms. A TRINOMIAL has three terms. A POLYNOMIAL with more than three terms is just called a POLYNOMIAL. 4.3 S KM & PP

Polynomial Classification by the Degree of the Leading Term Once a POLYNOMIAL is arranged in STANDARD FORM, the term with the largest exponent on the variable is called the LEADING TERM. The exponent of the variable in the leading term is the DEGREE of the POLYNOMIAL. 4.3 S KM & PP

Polynomial Classification: Degree Names Degree 0 is called CONSTANT. Degree 1 is called LINEAR. Degree 2 is called QUADRATIC. Degree 3 is called CUBIC. Degree 4 is called 4th degree, and so on for the higher powers. 4.3 S KM & PP

Monomial or Term Examples of a Monomial: a=2 and n = 7 a=-4 and n = 1 KM & PP

Monomial or Term In words: The coefficient is 2 and the degree is 7. KM & PP

Polynomial Example: -3x+10 -3x+10 is a binomial because it has two terms. -3x+10 is the same as -3x1+10x0 is the “linear” term. The coefficient is -3. The degree is 1. is the “constant” term The degree is 0. The coefficient is 10. 4.3 S KM & PP

Polynomial Example: 5x2 – 3x + 10 5x2 – 3x + 10 is a trinomial because it has three terms. is the “quadratic” term. The degree is 2. The coefficient is 5. is the “linear” term. The degree is 1. The coefficient is -3. is the “constant” term. The degree is 0. The coefficient is 10. 4.3 S KM & PP

Polynomial Example: 7x3 + 5x2 – 3x + 10 7x3 + 5x2 – 3x + 10 is a polynomial. is the “cubic” term. The degree is 3. The coefficient is 7. is the “quadratic” term. The degree is 2. The coefficient is 5. is the “linear” term. The degree is 1. The coefficient is -3. is the “constant” term. The degree is 0. The coefficient is 10. 4.3 S KM & PP

Let’s Arrange and Analyze this Polynomial Degree Quadratic (highest power is 2) Standard Form - Decending Powers Leading Coefficient is -5 Linear Term (variable to the first power) Constant term (variable to the zero power) 4.3 S KM & PP

Simplifying a Polynomial combine "like" terms quadratic terms linear terms constant terms write in Standard Form 4.3 S KM & PP

Simplify Another Polynomial combine "like" terms quadratic terms linear terms constant terms write in Standard Form rewrite without the "missing" term (coefficient is zero) 4.3 S KM & PP

How to EVALUATE a POLYNOMIAL To EVALUATE a polynomial we substitute for the variable with the given number or expression. (Replace the variable with parentheses and substitute. 4.3 S KM & PP

EVALUATE -5x2 + 3x – 1 for x = 2 4.3 S KM & PP

EVALUATE -5x2 + 3x – 1 for x = 0 4.3 S KM & PP

EVALUATE -5x2 + 3x – 1 for x = -2 4.3 S KM & PP

is called "delta x" Evaluate 3x -2 for x = x+Δx One for Calculus? KM & PP

That’s All for Now! 4.3 S KM & PP