A polynomial is an expression containing any number of variables and constants. The variables are combined by adding, subtracting, and multiplying. The variables themselves can be raised to a positive whole-number power.
A monomial is the product of any number of variables, each raised to any positive whole-number power. Thus, monomials do not involve addition or subtraction. Monomials may be multiplied by a constant.
These are all monomials:
A binomial is the sum of two monomials.
A trinomial is the sum of three monomials (or a binomial and monomial).
Simplifying a polynomial (or "collecting like terms") is the process of reducing a polynomial to its shortest form. The number before a term is the term's coefficient. Add or subtract the coefficients of terms that have the same combination of variables. That is, add or subtract the coefficients of like terms.
To make communication of monomials easier, we oftentimes use the convention of writing the variables of each term in alphabetical order, and we use exponential notation so that in each term each letter appears only once. We like to put the number (the numerical coefficient) at the beginning of the term.
If the terms are each expressed in such a manner, we can quickly identify like terms. Two terms are "like" if, when you cover up the coefficient of each term, the rest of the terms are identical to one another.
We cannot combine "unlike" terms. That is, we leave them alone.
A polynomial must be simplified before it can be classified as a monomial/binomial/trinomial/polynomial.
We can talk about the degree of a term, or the degree of a polynomial for a certain variable. In this context, I am using "polynomial" to include monomials, binomials, and trinomials.
Most of the time, we talk about degree for polynomials that only contain one variable. In this setting, the degree of a single term is the exponent for the variable in that term. For example:
- The degree of is zero.
- The degree of is one.
- The degree of is five.
For a polynomial of a single variable, the degree of the polynomial is the largest exponent that appears on that variable. The degree of is three.
To find the leading coefficient of a polynomial, identify the numerical coefficient of the term having the largest degree. The leading coefficient of is negative two.
To make things easier, we oftentimes like to write polynomials either in ascending or descending order. Ascending order means that the degrees of the terms ascend (get bigger)
as you go from left to right, and descending order means that the degrees of the terms descend (get smaller) as you go from left to right. When we write
in descending order, we get .
If a polynomial is given in descending order, then the degree of the polynomial is the degree of the first term, and the leading coefficient is the number out front.
Factoring polynomials deals with picking out the common factor. Just like before when we factored a real number we apply the same idea to binomials, trinomials, and other polynomials.
We picked out the common variable in the trinomial. If all of these would have been different variables there would be no common variables to factor.
Just like when we factored out the common variables we can also factor out coefficients with the variables.
When there is no common term that travels throughout the polynomial you can factor by grouping. When factoring by grouping you need to split up the polynomial into two binomials.
From here we group the first binomial and factor out the common variable.
From there we group the second binomial from the polynomial.
This can also be used to group polynomials with different variables.
By remembering the foil method we can move backwards and solve this problem. We know that we multiply "first" so there must not be any coefficients in the first two variables of the binomials.
From there we go to "inside". We see that the first term is multiplied by the last term in the second binomial. So we get
Now go to the "outside" and we see . Since we have the first term of the second binomial then we can see that the last term in the first binomial is . Therefore, we get
This doesn't always work because the middle terms often end up having the same common variables and get simplified. It's helpful to start with the first terms and then go to the last terms. If you can get these two things you should be able to find the middle term/s.