A polynomial is an algebraic expression that includes addition, subtraction, multiplication, and whole number exponents, such as:

4x3 – 3x2 + 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. 4x3 – 3x2 + 7x + 5

has four terms.

How many terms are in each of these polynomials? x2 – 2x + 3

4n5 + 2n4 + 3n3 – 2n2

5x2y5 + 2x3y6

How many terms are in each of these polynomials? x2 – 2x + 3 3

4n5 + 2n4 + 3n3 – 2n2 4

5x2y5 + 2x3y6 2

In the polynomial

4x3 + 2y2 – 5z + 2 How many terms are there? What is the coefficient on the 1st

term? What is the coefficient on the 3rd

term? What is the base on the 2nd term? Which term is a constant?

In the polynomial

4x3 + 2y2 – 5z + 2 How many terms are there? 4 What is the coefficient on the 1st

term? 4 What is the coefficient on the 3rd

term? -5 What is the base on the 2nd term? y Which term is a constant? 4th

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 5x2y4 + 2xy3 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 5x2y4 + 2xy3, the degree of the 1st term is 6 and the degree of the 2nd term is 4.

What is the degree of each term of this polynomial?

5x2y2 – 2x5 + 4x7y3 – 7x + 5

What is the degree of each term of this polynomial?

5x2y2 – 2x5 + 4x7y3 – 7x + 5 4 5 10 1 0

The degree of a whole polynomial is the largest of the degrees of its terms.

So, the degree of 5x2y4 + 2xy3 is 6.

The degree of 5x2y2 – 2x5 + 4x7y3 – 7x + 5is 10.

What is the degree of 5n5p3 + 2n2p7 – 4n4p3 + 5np6?

What is the degree of 5n5p3 + 2n2p7 – 4n4p3 + 5np6? 8 9 7 7The overall degree is 9.

Special types of polynomials: Quadratic

One variableDegree = 2

CubicOne variableDegree = 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.

(5x2 + 3x – 2) + (3x2 – 7x + 8)

= 8x2 – 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:

(6x2y4 + 2x3y + 3x2y2)+ (5x3y – 2x2y4 + 2x2y2)

Simplify:

(6x2y4 + 2x3y + 3x2y2)+ (5x3y – 2x2y4 + 2x2y2)

= 4x2y4 + 7x3y + 5x2y2

Simplify:

(y3 – 2y2 + 3y + 1) + (3y3 + 2y2 – 5y + 2)

Simplify:

(y3 – 2y2 + 3y + 1) + (3y3 + 2y2 – 5y + 2)

4y3 – 2y + 3

Simplify:

(7n4 + 8n3 – 5n2) – (2n4 + 3n3 + n2)

Simplify:

(7n4 + 8n3 – 5n2) – (2n4 + 3n3 + n2)

5n4 + 5n3 – 6n2

Simplify:

(5x2 – 3x + 5) – (2x2 + 4x – 1)

Simplify:

(5x2 – 3x + 5) – (2x2 + 4x – 1)

3x2 – 7x + 6

Simplify:

(7x2 + 2y2) – (4x2 + 2xy – 3y2)

Simplify:

(7x2 + 2y2) – (4x2 + 2xy – 3y2)

3x2 – 2xy + 5y2