Polynomial Operations
Polynomials
An algebraic expression consisting of one or more variables and one or more terms.
The exponent(s) on the variables must be positive integers only.
Monomial – a one-term polynomial ( 3xy )
Binomial – a two-term polynomial ( 3 xy 5 x 2 )
Trinomial – a three-term polynomial ( 3xy 5 x 2 2 y )
Degree of a term is the sum of the exponents of the variables in that term
Example:
( 3x 2 y 3 is of degree 5)
Degree of the polynomial is the degree of the term with the highest degree
Example:
( xy 2 y 2 x 2 y 2 )
has terms of degrees (2, 2, 4) so the polynomial has
an overall degree of 4
Like terms Like terms are terms that contain the same variables raised to the same powers.
2xy 2 and 3xy 2 are like terms
Example:
7xy and 10x 2 y 2 are unlike terms
Adding and/or Subtracting Polynomials
Add or subtract the like terms only.
Example: (3xy + 5x2 + 2y) – (7x2 + 2xy – 5y + 7xyz)
= 3xy – 2xy + 5x2 – 7x2 + 2y + 5y – 7xyz
= xy – 2x2 + 7y – 7xyz
Multiplying Two Binomials
(a + b)(c + d)
FOIL Method:
a c
First =
Outer =
a d
Inner =
b c
Last =
b d
Therefore, F + O + I + L = ac + ad + bc + bd
Example:
(x + 3)(x + 5) = x2 + 5x + 3x + 15 = x2 + 8x + 15
Page 1
Multiplying Polynomials - General
When multiplying polynomials, multiply each term in the first polynomial by each term in the second
polynomial.
Example :
(x 2
3xy)(4 x xy 7 y 2 )
( x 2 )(4 x) ( x 2 )( xy) ( x 2 )(7 y 2 ) (3xy)(4 x) (3xy)( xy) (3xy)(7 y 2 )
4x 3
x 3 y 7 x 2 y 2 12x 2 y 3x 2 y 2
21xy 3
4x 3
x 3 y 10x 2 y 2 12x 2 y 21xy 3
Special Patterns
Trinomial Squares
( a b) 2
(a b)(a b) a 2 2ab b 2
( a b) 2
(a b)(a b) a 2 2ab b 2
a 2 b2
Difference of Two Squares
(a b)(a b)
(a+b)2
(a–b)2
NOTE:
a 2 + b2
a2 – b2
ex.
(1+2)2
12 + 2 2
ex.
(5–3)2
52 – 32
Dividing Monomials
x
m
xm
x
n
xn
xm
xn
xm
xn
xm
n
when m > n
1
x
when m < n
n m
Dividing a Polynomial by a Monomial
(a + b)
c=
a b
a
b
= +
c
c
c
Dividing a Polynomial by a Polynomial
Use the long division algorithm (Divide, Multiply, Subtract, Bring down; repeat as needed)
Example: (4x3 – 5x2 + 3x + 6)
4x 2
x 2 4x
3x
5x
3
8x )
(4x
24
9
3
(x – 2)
x 2
3x 6
2
2
3x
2
(3x
2
3x
6x)
9x 6
(9x 18)
24
LSC-Montgomery Learning Center: Polynomial Operations
Last Updated April 13, 2011
2