A polynomial is an algebraic expression that includes addition, subtraction, multiplication, and whole number exponents, such as: 4x 3 – 3x 2 + 7x + 5

A polynomial can not include  roots or fractional exponents  fractions with the variable on the bottom (or negative exponents)  variables in the exponent

“Polynomial” means “many parts”.  The parts of a polynomial are called terms.  Each + or – starts a new term.  4x 3 – 3x 2 + 7x + 5 has four terms.

How many terms are in each of these polynomials?  x 2 – 2x + 3  4n 5 + 2n 4 + 3n 3 – 2n 2  5x 2 y 5 + 2x 3 y 6

How many terms are in each of these polynomials?  x 2 – 2x  4n 5 + 2n 4 + 3n 3 – 2n 2 4  5x 2 y 5 + 2x 3 y 6 2

In the polynomial 4x 3 + 2y 2 – 5z + 2  How many terms are there?  What is the coefficient on the 1 st term?  What is the coefficient on the 3 rd term?  What is the base on the 2 nd term?  Which term is a constant?

In the polynomial 4x 3 + 2y 2 – 5z + 2  How many terms are there? 4  What is the coefficient on the 1 st term? 4  What is the coefficient on the 3 rd term? -5  What is the base on the 2 nd term? y  Which term is a constant? 4 th

We often classify polynomials by the number of terms they have. In particular …  Monomial  Binomial  Trinomial

We often classify polynomials by the number of terms they have. In particular …  Monomial=1 term  Binomial =2 terms  Trinomial=3 terms

Your book will sometimes ask you to write polynomials in standard form. This just means re-arranging the terms so the exponents count down from left to right.

You can have more than one variable in a polynomial. For instance 5x 2 y 4 + 2xy 3 is a binomial where each term has 2 variables.

The degree of a term tells how big that term is. The degree is the sum of the exponents in a term. In 5x 2 y 4 + 2xy 3, the degree of the 1 st term is 6 and the degree of the 2 nd term is 4.

What is the degree of each term of this polynomial? 5x 2 y 2 – 2x 5 + 4x 7 y 3 – 7x + 5

What is the degree of each term of this polynomial? 5x 2 y 2 – 2x 5 + 4x 7 y 3 – 7x

The degree of a whole polynomial is the largest of the degrees of its terms. So, the degree of 5x 2 y 4 + 2xy 3 is 6. The degree of 5x 2 y 2 – 2x 5 + 4x 7 y 3 – 7x + 5 is 10.

What is the degree of 5n 5 p 3 + 2n 2 p 7 – 4n 4 p 3 + 5np 6 ?

What is the degree of 5n 5 p 3 + 2n 2 p 7 – 4n 4 p 3 + 5np 6 ? The overall degree is 9.

Special types of polynomials:  Quadratic One variable Degree = 2  Cubic One variable Degree = 3

It’s easy to add or subtract polynomials.  Just combine like terms.  When you do that, you add the coefficients, and leave the variables and coefficients alone.

(5x 2 + 3x – 2) + (3x 2 – 7x + 8) =8x 2 – 4x + 6

The variables and exponents act like a label for the terms, which is why they don’t change when you add or subtract.

Simplify: (6x 2 y 4 + 2x 3 y + 3x 2 y 2 ) + (5x 3 y – 2x 2 y 4 + 2x 2 y 2 )

Simplify: (6x 2 y 4 + 2x 3 y + 3x 2 y 2 ) + (5x 3 y – 2x 2 y 4 + 2x 2 y 2 ) = 4x 2 y 4 + 7x 3 y + 5x 2 y 2

Simplify: (y 3 – 2y 2 + 3y + 1) + (3y 3 + 2y 2 – 5y + 2)

Simplify: (y 3 – 2y 2 + 3y + 1) + (3y 3 + 2y 2 – 5y + 2) 4y 3 – 2y + 3

Simplify: (7n 4 + 8n 3 – 5n 2 ) – (2n 4 + 3n 3 + n 2 )

Simplify: (7n 4 + 8n 3 – 5n 2 ) – (2n 4 + 3n 3 + n 2 ) 5n 4 + 5n 3 – 6n 2

Simplify: (5x 2 – 3x + 5) – (2x 2 + 4x – 1)

Simplify: (5x 2 – 3x + 5) – (2x 2 + 4x – 1) 3x 2 – 7x + 6

Simplify: (7x 2 + 2y 2 ) – (4x 2 + 2xy – 3y 2 )

Simplify: (7x 2 + 2y 2 ) – (4x 2 + 2xy – 3y 2 ) 3x 2 – 2xy + 5y 2