lesson 5

# Polynomials

A polynomial is an expression involving constants, variables
with natural number exponents, the four basic operations of +, -, . and / and no
variables in denominators.

Rules for Addition and Subtraction of Polynomials

1. Two terms cannot be added (or subtracted) unless they consist of the same variables to the same exponents. These are called similar (or like) terms.
2. To add (or subtract) two similar terms, add (or subtract) the coefficients, leaving the variables unchanged.

Example: Perform the operations and simplify:

\eqalign{ & {\rm{(2}}{{\rm{x}}^{\rm{2}}}{\rm{ + 3xy + }}{{\rm{y}}^{\rm{2}}}{\rm{) – (2}}{{\rm{x}}^{\rm{3}}}{\rm{ – 4xy + }}{{\rm{y}}^{\rm{2}}}{\rm{) + (4}}{{\rm{x}}^{\rm{3}}}{\rm{ + xy – 5}}{{\rm{x}}^{\rm{2}}}{\rm{)}} \cr & {\rm{2}}{{\rm{x}}^{\rm{2}}}{\rm{ + 3xy + }}{{\rm{y}}^{\rm{2}}}{\rm{ – 2}}{{\rm{x}}^{\rm{3}}}{\rm{ + 4xy – }}{{\rm{y}}^{\rm{2}}}{\rm{ + 4}}{{\rm{x}}^{\rm{3}}}{\rm{ + xy – 5}}{{\rm{x}}^{\rm{2}}} \cr & {\rm{2}}{{\rm{x}}^{\rm{3}}}{\rm{ – 3}}{{\rm{x}}^{\rm{2}}}{\rm{ + 8xy}} \cr}

To multiply polynomials, we use the distributive property and the rules of exponents.

Multiply every term of the first polynomial by every term of the second
polynomial.

Example: Perform the operations and simplify:

\eqalign{ & {\rm{(2x + 3y)(x + 2y)}} \cr & = {\rm{(2x + 3y)x + (2x + 3y)2y}} \cr & {\rm{ = 2}}{{\rm{x}}^{\rm{2}}}{\rm{ + 3xy + 4xy + 6}}{{\rm{y}}^{\rm{2}}} \cr & {\rm{ = 2}}{{\rm{x}}^{\rm{2}}}{\rm{ + 7xy + 6}}{{\rm{y}}^{\rm{2}}} \cr}

To factor an expression means to write it as a product.

The greatest common factor, or GCF, of two or more terms is the largest expression that is a factor of all of the terms.

Example: Factor the greatest common factor out of the expression:

$${\rm{14}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{3}}}{\rm{z – 7}}{{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{2}}}{\rm{ + 2l}}{{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{3}}}{{\rm{z}}^{\rm{3}}}{\rm{ – 28}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{4}}}{{\rm{z}}^{\rm{2}}}$$

First we factor each term completely, then we find the factors that are common to all of the terms

\eqalign{ & {\rm{14}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{3}}}{\rm{z – 7}}{{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{2}}}{\rm{ + 2l}}{{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{3}}}{{\rm{z}}^{\rm{3}}}{\rm{ – 28}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{4}}}{{\rm{z}}^{\rm{2}}} \cr & {\rm{14}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{3}}}{\rm{z}} = 2*7*{\rm{x}}*{\rm{x}}*{\rm{y}}*{\rm{y}}*{\rm{y}}*{\rm{z}} \cr & {\rm{7}}{{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{2}}}{\rm{ = }}7*{\rm{x}}*{\rm{x}}*{\rm{x}}*{\rm{y}}*{\rm{y}}*{\rm{z}}*{\rm{z}} \cr & {\rm{21}}{{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{3}}}{{\rm{z}}^{\rm{3}}} = 3*7*{\rm{x}}*{\rm{x}}*{\rm{x}}*{\rm{y}}*{\rm{y}}*{\rm{y}}*{\rm{z}}*{\rm{z}}*{\rm{z}} \cr & {\rm{28}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{4}}}{{\rm{z}}^{\rm{2}}} = 2*2*7*{\rm{x}}*{\rm{x}}*{\rm{y}}*{\rm{y}}*{\rm{y}}*{\rm{y}}*{\rm{z}}*{\rm{z}} \cr & {\rm{7}}{{\rm{x}}^{\rm{2}}}{{\rm{y}}^{\rm{2}}}{\rm{z(2y – xz + 3xy}}{{\rm{z}}^{\rm{2}}}{\rm{ – 4}}{{\rm{y}}^{\rm{2}}}{\rm{z)}} \cr}