POLYNOMIALS
Polynomials are
expressions with more than one term, such as
3x + 2y
+ 8x - 9
We call an
expression with two terms, like 3x + 2y, a "binomial."
MULTIPLYING
POLYNOMIALS
To multiply two
algebraic expressions, multiply every term in the first expression by every
term in the other expression.
Example: 3x (5x + 2y - z)
=(3x)(5x) +
(3x)(2y) - (3x)(z)
=15
+ 6xy - 3xz
Example: (a + b)(c + d) = ab + ad + bc + bd
When we multiply
two binomials we use the FOIL method.
FOIL stands for "First Outer Inner Last" and is a way to
remember to multiply the
First terms (a x
c)
Outer terms (a x
d)
Inner terms (b x
c)
Last terms (b x
d)
Example: (a + 2)(a - 5) = 
- 5a + 2a - 10 = 
- 3a -10
Example: 
= (x + y)(x + y)
= 
+ xy + yx + 
= 
+ xy + xy + 
= 
+ 2xy + 
Caution: 
is NOT the same as 
+ 
(This is a very
common error!)
VERY IMPORTANT
EQUATIONS
The following
relationships are very important and should be memorized.
I. 
= 
+ 2xy + 
II. 
= 
- 2xy + 
III. (x + y)(x - y) = 
- 
(This is
called the "difference of squares" since 
and y2 are
squares and 
- 
is the difference of
two squares.)
Example: 
=
+ 2(3x)+ 
= 
+ 6x + 9
Example: (a + 2)(a - 2)= 
- 4
FACTORING
POLYNOMIALS
To factor
polynomials we do the reverse of multiplication.
Example: Factor 15
- 35xy
Solution: Look for a common numerical factor and look
for a common factor with a variable.
Since 15 and 35 are both divisible by 5, 5 is a common factor of both
terms. Since 
and xy are both
divisible by x, x is a common factor of both terms.
Therefore, 15
- 35xy = 5x(3x - 7y)
Example: Factor 
- 25
Solution: 
and 25 are both
squares, so 
- 25 is the
difference of squares. Therefore 
- 25 = (x + 5)(x -
5).
SOLVING
EQUATIONS WITH POLYNOMIALS
If A and B are
two numbers and if AB = 0, then either A = 0 or B = 0. To solve equations where polynomials are
equal to zero, we factor the polynomial and set each factor equal to 0 and
solve.
Example: 
- 6x = 0
Since 
- 6x = x(x - 6) we can find values of x which satisfy the
equation by setting x = 0 and x - 6 = 0.
Therefore, x = 0 or +6.
Check: 3
- 6(0) = 0 + 0 = 0
- 6(6) = 36 - 36 = 0.
Example: 
- 25 = 0
We wish to find
values of x that will make the equation a true statement. The trick is to factor 
- 25 and set each
factor equal to 0.
- 25 = 0
(x + 5)(x - 5) =
0
Therefore, x + 5
= 0 means that x = -5
and x - 5 = 0
means that x = +5.
Therefore, there
are two solutions, x = 5, and x = -5.