1.3 – Algebraic Expressions – Factoring
Section Notes
Polynomials
A polynomial of variable is an expression of the form
where the coefficients are real numbers, and is nonnegative. If , then the polynomial has degree of . Each monomial of the form that make up the polynomial are called the terms of the polynomial.
More simply stated, a Polynomial is the collection (sum or difference) of one or more terms. The word polynomial, when broken down means many (poly) terms (nomial). The following are examples of polynomials:
Notice that the number 4, which is a term, can also be considered a polynomial. It is a polynomial of one term.
A Monomial is a polynomial containing only one term. Examples of monomials are:
A Binomial is a polynomial containing two terms. Examples of binomials are:
A Trinomial is a polynomial containing three terms. Examples of trinomials are:
Polynomials with more than three terms are generally just referred to simply as polynomials.
The largest term exponent is called the degree of the polynomial, and the coefficient of the largest degree term is called the leading coefficient.
Two terms are considered to be like terms if the have the same variable(s) that are raised to the same power(s).
For example, and are like terms.
If a polynomial has two like terms, it can be simplified by combining them. Consider the expression . Since multiplication is repeated addition, we can express as and can be expressed as . We can then rewrite the expression:
We can then recombine this as multiplication, writing:
So we see that we can combine terms by adding them together. This is how we will treat the addition of polynomials. When adding polynomials, combine like terms, and write in descending order.
To add the polynomials and , we simply write a sum between the two polynomials:
And the parentheses can be removed:
The resulting polynomial can be reordered, and like terms can be added together, ensuring that the polynomial is written in descending order:
Subtracting polynomials requires some attention to details. In polynomial addition, we simply removed the parentheses and combined. In polynomial subtraction, one must distribute the negative through the parentheses before combining.
To subtract the polynomial from the polynomial first write the subtraction,
then distribute the negative through the second polynomial,
and then combine like terms:
Multiplying Polynomials makes use of the distribution property.
Multiply We can then use the distributive property again, to take the factors on the right through each : And then multiplying to simplify each term: At this point, we will want to simplify further by adding like terms: 

When we have two binomials, we might make use of the FOILFirst, Outer, Inner, Last, and denotes the order in which one might multiply elements of the binomials.
For larger polynomial multiplication, there are no cute acronyms, and the distribution method is simply employed:
Multiply 

Using the distributive property: And simplifying by adding like terms: 
Perfect Squares 

A Perfect Square is the result of a binomial being multiplied by itself:

The Product of the Sum and Difference of Two Terms 

The Product of the Sum and Difference of Two Terms simplifies to the difference of the squares of the terms: 
Sum and Difference of two cubes 

Similar to the creation of the difference of two squares above, the sum and difference of two cubes are created by the following:

In summary:
Special Product Formulas
These formulas can be viewed in the opposite direction as a guide to factoring special cases:
Special Factoring Formulas