Polynomials Notes
Polynomials are special types of algebraic expressions.
A polynomial is in simplest form when all like terms are collected and terms appear in
descending order of the variable.
A term of a polynomial is a number, variable, or the product of a number and a variable.
The terms of a polynomial are the parts separated by an add or subtract sign.
The coefficients are the numerical factors of any term of a polynomial.
A monomial is a polynomial containing one term.
A binomial is a polynomial containing two terms.
A trinomial is a polynomial containing three terms.
The degree of a term of a polynomial with one variable is the exponent on the variable.
The degree of a polynomial in one variable is the highest exponent of the variable.
To find the degree of term with more than one variable add the exponents of all variables in
that term.
The degree of a polynomial with more than one variable is the highest or largest sum of the
exponents in any one term of a the polynomial.
The constant term has a degree of zero, and is any number, other than zero.
For example 7x0  71  7
Now rewrite the polynomial 2x 2  7x 5  9x  8x 6  x 3  2 in descending order and fill in the
following chart.
Rewriting the polynomial in descending order  8x 6  7x 5  x 3  2x 2  9x  2
Terms
Coefficients
Degree of each
term
Degree of the
polynomial
 8x 6
8
6
6
 7x 5
7
5
 x3
1
3
2x 2
2
2
9x
2
9
1
2
0
Now rewrite the polynomial 5x  x 2  5x 7  9  3x 5  8x 4 in descending order and fill in the
following chart.
Rewriting the polynomial in descending order 5x 7  3x 5  8x 4  x 2  5x  9
Terms
Coefficients
Degree of each
term
Degree of the
polynomial
5x 7
5
7
3x 5
3
5
8x 4
8
4
 x2
1
2
5x
9
5
1
9
0
7
Now rewrite the polynomial  11x 4  3x 8  11x  3x 3  4x 7  2 in descending order and fill in
the following chart.
Rewriting the polynomial in descending order______________________________________
Terms
Coefficients
Degree of each
term
Degree of the
polynomial
Now rewrite the polynomial 5x 3  3x 2  7x  4x 4  x 5  7 in descending order and fill in the
following chart.
Rewriting the polynomial in descending order______________________________________
Terms
Coefficients
Degree of each
term
Degree of the
polynomial
In order to add polynomial like terms most be collected. Like terms have exactly the same
combination of variables with the same exponents.
7x 3 y 4
These are like terms.
These are unlike terms.
 11x3 y4
7x 3 y 4
 11x 4 y3
To add polynomials
1.
2.
3.
4.
Combine like terms in descending order of the variable (if only one variable).
Add the coefficients only, do not add the exponents on the variables.
Polynomials can be added horizontally or vertically.
Can use the commutative or associative properties to rearrange terms.
To subtract polynomials
1.
2.
3.
4.
5.
Remove parentheses and change the subtract sign to an add sign, then change all
the following signs on the second polynomial to their opposite.
Collect like terms in descending order of the variable (if only one variable).
Add the coefficients only, do not add the exponents on the variables.
Polynomials can be subtracted horizontally or vertically.
If subtracting polynomials vertically, copy the problem, then copy the problem again,
changing all the signs on the second polynomial to their opposite, then follow the
rules above.
Sometimes a subtract problem is written in the following form.
Subtract 8 from 12. It is much easier to rewrite this problem in the form 12  8  4
Subtract 3 from 15. It is much easier to rewrite this problem in the form 15  3  12
Subtract 6x 2  4x  3 from 3x 2  11x  4 .
In order to subtract this problem we should rewrite it with parentheses. The polynomial
written after the word from is written first. Remove the parentheses and change the subtract
sign to an add sign, then change all the following signs on the second polynomial to their
opposite. Collect like terms in descending order of the variable (if only one variable). Add
the coefficients only, do not add the exponents on the variables.
Subtract 6x 2  4x  3 from 3x 2  11x  4 .
3x
2

 11x  4 
6x
2
 4x  3

3x 2  11x  4   6x 2  4x  3
 3x 2  15x  7
Let’s look at another example.
Subtract  2x2  13x  2 from 6x2  3x  5 .
6x
2
 3x  5

6x  3x  5 
2
 2x

2
 13x  2

2x  13x  2
2
8x  16x  7
2
Sometimes a problem contains three polynomials to add or subtract. In the problem below
perform the addition of the first two polynomials and then go through and change the signs
on the third polynomial, and collect like terms in descending order.
  3x
2
 
 7x  5  9x2  8x  11
    2x

2
 3x  7
2
 8x  12
12x2  15x  16  2x2  3x  7
10x2  12x  9
Let’s try another example.
  2x
2
 
 7x  9  11x2  4x  1
    5x

2x2  7x  9  11x2  4x  1  5x2  8x  12
 14x2  5x  4
Here is another example.
  5x
2
 
 2x  3  7x2  11x  9
    3x
2
 6x  4

5x2  2x  3  7x2  11x  9  3x2  6x  4
 5x2  3x  10
Try this example.
  3x
2
 
 6x  7  9x2  3x  12
    7x
2
 11x  2
