Mathematics: Polynomial Algebra

Polynomials
Polynomials are made up of monomials and polynomials. A monomial is a single variable or product of constants and variables, such as x, 2x, or
 . There will never be addition or subtraction symbols in a monomial. Like monomials have like variables, but they may have different coefficients. Polynomials are algebraic expressions which use addition and subtraction to combine two or more monomials. Two terms make a binomial, three terms make a trinomial, etc. The degree of a monomial is the sum of the exponents of the variables. The degree of a polynomial is the highest degree of any individual term.

Simplifying Polynomials
Simplifying polynomials requires combining like terms. The like terms in a polynomial expression are those that have the same variable raised to the same power. It is often helpful to connect the like terms with arrows or lines in order to separate them from the other monomials. Once you have determined the like terms, you can rearrange the polynomial by placing them together. Remember to include the sign that is in front of each term. Once the like terms are placed together, you can apply each operation and simplify.
When adding and subtracting polynomials, only add and subtract the coefficient, or the number part; the variable and exponent stay the same.

The FOIL Method
In general, multiplying polynomials is done by multiplying each term in one polynomial by each term in the other and adding the results. In the specific case for multiplying binomials, there is useful acronym, FOIL, that can help you make sure to cover each combination of terms. The FOIL method for  would be:

F
Multiply the first terms of each binomial -
-

O
Multiply the outer terms


I
Multiply the inner terms


L
Multiply the last terms of each binomial



Then add up the result of each and combine like terms:
.
For example, using the FOIL method on binomials

and
:

This results in:

Combine like terms:


Dividing Polynomials
To divide polynomials, set up a long division problem, dividing a polynomial by either a monomial or another polynomial of equal or lesser degree.
When dividing by a monomial, divide each term of the polynomial by the monomial.
When dividing by a polynomial, begin by arranging the terms of each polynomial in order of one variable. You may arrange in ascending or descending order, but be consistent with both polynomials. To get the first term of the quotient, divide the first term of the dividend by the first term of the divisor. Multiply the first term of the quotient by the entire divisor and subtract that product from the dividend. Repeat for the second and successive terms until you either get a remainder of zero or a remainder whose degree is less than the degree of the divisor. If the quotient has a remainder, write the answer as a mixed expression in the form:


For example, we can evaluate the following expression in the same way as long division:



















45


When factoring a polynomial, first check for a common monomial factor, that is look to see if each coefficient has a common factor or if each term has an   in it. If the factor is a trinomial but not a perfect trinomial square, look for a factorable form, such as one of these:
 

 

For factors with four terms, look for groups to factor. Once you have found the factors, write the original polynomial as the product of all the factors. Make sure all of the polynomial factors are prime. Monomial factors may be prime or composite. Check your work by multiplying the factors to make sure you get the original polynomial.

Below are patterns of some special products to remember to help make factoring easier:
 

Perfect trinomial squares:

or


· Difference between two squares:

· Sum of two cubes:

- Note: the second factor is not the same as a perfect trinomial square, so do not try to factor it further.

· Difference between two cubes:

o  Again, the second factor is not the same as a perfect trinomial square.

·     Perfect cubes:

and


Rational expressions
Rational expressions are fractions with polynomials in both the numerator and the denominator; the value of the polynomial in the denominator cannot be equal to zero. Be sure to keep track of values that make the denominator of the original expression zero as the final result inherits the same restrictions. For example, a denominator of 
indicates that the expression is not defined when 
and as such, regardless of any operations done to the expression, it remains undefined there.

To add or subtract rational expressions, first find the common denominator, then rewrite each fraction as an equivalent fraction with the common denominator. Finally, add or subtract the numerators to get the numerator of the answer, and keep the common denominator as the denominator of the answer.
When multiplying rational expressions factor each polynomial and cancel like factors (a factor which appears in both the numerator and the denominator). Then, multiply all remaining factors in the numerator to get the numerator of the product, and multiply the remaining factors in the denominator to get the denominator of the product. Remember: cancel entire factors, not individual terms.
To divide rational expressions, take the reciprocal of the divisor (the rational expression you are dividing by) and multiply by the dividend.

Simplifying Rational Expressions
To simplify a rational expression, factor the numerator and denominator completely. Factors that are the same and appear in the numerator and denominator have a ratio of 1. For example, look at the following expression:

The denominator,
, is a difference of squares. It can be factored as
. The factor

and the numerator

are opposites and have a ratio of –1. Rewrite the numerator as
. So, the rational expression can be simplified as follows:

Note that since the original expression is only defined for
, the simplified expression has the same restrictions.

Problems:

P1. Expand the following polynomials:
(a)

(b)

(c)


P2. Evaluate the following rational expressions:
(a)

(b)


P1. (a) Apply the FOIL method and the distributive property of multiplication:

(b) Note the difference of squares form:

(c) Multiply each pair of monomials and combine like terms:




P2. (a) Rather than trying to factor the fourth-degree polynomial, we can use long division:














0

Note that since the original expression is only defined for
, the The denominator,
, is a difference of squares. It can be factored as
. The numerator,
, is a perfect square. It can be factored as
. So, the
expression is only defined for