Polynomials
A polynomial is an expression that has two or more terms each
separated by a + or – sign. If the
expression has only one term it is called a monomial. If the expression
has two terms it is called a
binomial . If the expression has three terms it is called a Trinomial .
The degree of the polynomial is
determined by the highest power or exponent in any of the terms
a constant is a zero
degree monomial 5 |
a first degree
monomial−2x |
a second degree
binomial
 |
a second degree
binomial
 |
a second degree
trinomial
 |
a second degree
trinomial
 |
a third degree
trinomial
 |
a third degree
trinomial
 |
A polynomial is an expression with several terms. Like Terms are terms
that can be combined by
addition or subtraction to form a single term. For terms to be considered like
terms every term must
have the exact same variables (letters) and each variable must have the
exact same exponent
as all of the other terms. The coefficients are not used to determine if
the terms are alike.
Example of terms that are like terms include:
5x^2 and 3x^2
are like terms
because
the x^2 in each term
are the same letters to
the exact same powers |
−4y^3 and y^3
are like terms
because
the y^3 in each term
are the same letters to
the exact same powers |
2x2y and − 5x2y
are like terms
because
the x2y in each term
are the same letters to
the exact same powers |
Example of terms that are NOT like terms include:
5x^3 and 3x^2
are not like terms because
the x^3 and x^2 are not
to the same power |
−4x^2 and y^2
are not like terms because
the x and y terms are not
the exact same variables |
2xy2 and −5x2y
are not like terms because the
xy2 and x2y terms have the
same letters but they do not
have the exact same exponents |
Combining Like Terms
Combining like terms involves determining the total of
the coefficients of all the like terms. If there
are three x terms in an expression like 3x + 2x + 5x then you can combine all
three of the x terms and
express that total with only one x term. This is done by adding or subtracting
the coefficients of the x
terms and using that number as the coefficient of the variable term that was the
like term.
| Example 1 |
Example 2 |
| 3x + 2x + 5x |
 |
To combine the like terms 3x + 2x
+ 5x
you combine the 3 + 2 + 5 to get 10
and use the common variable name x to get |
To combine the like terms
you combine the 2 + 4 −12 to get –6
and use the common variable name x to get |
 |
 |
To Combine Like Terms
1. Combine the coefficients of the like terms by adding
or subtracting the coefficients.
2. Using the number in step 1 as the coefficient of the variable term that was
the like term.
| Example 3 |
Example 4 |
Example 5 |
5x + 8x
combine the 5 +8
=13x |
4y − 9y
combine the 4 − 9
= −5y |
10xy − xy
combine the 10 −1
= 9xy |
| Example 6 |
Example 7 |
Example 8 |
combine the 3−1
 |
combine the 8 −12
 |
combine the 3+1
 |
| Example 9 |
Example 10 |
Example 11 |
combine the 2 − 9 + 3
 |
3xy + 8xy − xy
combine the 3+ 8 −1
=10xy |
2y − 7y + 5y
combine the 2 − 7 + 5
= 0 |
List the variables in any single term in
alphabetical order
If a single term has more then one variable
we list the letters in alphabetical order. This allows us to
more easily compare terms to see if they are alike. We do not write a term with
an x and y as both
xy and yx. It would be easy to think that they are not like terms.
When you put them in the correct
alphabetical order then it is clear they are like terms.
List all Polynomials In Descending Order
List the term with the highest power first
and then list the other terms in descending order of
their powers:
starts with the fourth power listed first
and then lists the terms with lower powers in order.
Simplify Polynomials with several
different terms
If a polynomial has several terms then all
the terms may not be like terms. For example, the
expression 3x^2 + 5x +12x^2 + 8x has like terms 5x and 8x and like
terms 3x^2 and 12x^2 but the four
terms are not all alike. The 5x and 8x are like terms and can be
replaced by 13x . The 3x^2 and 12x^2
are like terms and can be replaced by 15x^2 but you cannot combine the
15x^2 and the 13x . If that is
the case then combine the different kinds of like terms separately and list the
terms in descending
order based on the terms powers.
| Example 1 |
Example 2 |
Example 3 |
5x + 4x^2 + 3x + 2x^2
for x^2 combine the 4 + 2
for x combine the 5 + 3
= 6x^2 + 8x |
3x + 8 − 7x −2
for x combine the 3− 7
combine the constants 8 −2
= −4x + 6 |
3y −5y + 2y^2 − 9y^2
for y2 combine the 2 −9
for y combine the 3 −5
= −7y^2 −2y |
| Example 4 |
Example 5 |
Example 6 |
for x^2 combine the
for x combine the 5 − 9
= 2x^2 − 4 x |
for x^2 combine the -6 +2
for x combine the
= −4x^2 − 2x |
for y^2 combine the
for y combine the
= −3y^2 −2y |
