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VI FACTORING REVIEW
VOCABULARY REVIEW
coefficient - the number which precedes the variable in a monomial
example : -3xy - the coefficient is -3
monomial - an expression that is either a numeral, a variable or the product of a
numeral and one or more variables.
examples : 7 is a monomial - a numeral (also called a constant)
h is a monomial - a variable
1
c is a monomial - a product of a numeral and a variable
2
8x 2 y is a monomial - a product of a numeral and variables.
polynomial - an expression that is the sum of monomials
examples : 2 x 2  3xy  y 2 - is a polynomial which is called a trinomial (3 terms)
4 y 2 1 - is a polynomial which is called a binomial (2 terms)
prime number - an integer greater than one whose only factors are one and itself
2, 3, 5, 7, 11, 13 are examples of prime numbers
greatest common factor - the greatest number which is a factor of each of a
given set of numbers.
greatest common monomial factor - common factor that has the greatest
coefficient and the greatest degree in each variable
FACTORING GUIDELINES
A polynomial is said to be factored completely when it is expressed as the product
of prime polynomials and a monomial. The following are guidelines to help factor.
1) Factor out the greatest common monomial factor
2) Look for a difference of two squares
3) Look for a trinomial square
4) If a trinomial is not a square, look for a pair of binomial factors
5) If a polynomial has four or more terms, look for a way to group the terms in
pairs or in a group of three terms that is a binomial square
6) Make sure that each factor is prime. Check your factors by multiplying them.
Example 1. Factor 2a 2  4a  70 .
First, factor out the greatest common monomial factor
= 2(a 2  2a  35)
Then, find the binomial factors of the trinomial which is left. Consider the factors
of 35 which have a difference of 2. Since the sign of the middle (linear) term is
negative, this indicates that the sign of the larger factor of 35 should be negative.
 2(a  7)(a  5)
Check the answer by multiplying the factors.
2(a  7)(a  5)  2(a 2  7a  5a  35)
 2(a 2  2a  35)  2a 2  4a  70
Example 2. Factor 2 x3  7 x 2 15x
First factor out the greatest common monomial factor
 x( 2 x 2  7 x 15)
Then, find the binomial factors of the trinomial which is left. Since the
coefficient of the quadratic term (x2) is 2, this means that all combinations of the factors
of 2 must be considered as well as the factors of 15.
 x  2 x  3 ( x  5)
Check the answer by multiplying the factors.
x(2 x  3)( x  5)  x(2 x 2  10 x  3x  15)
 x(2 x 2  7 x 15)  2 x3  7 x 2 15 x
Example 3. Factor 25 y 2 100
First factor out the greatest common monomial factor
 25( y 2  4)
Then, since the remaining polynomial is two terms, check to see if it is the
difference of two squares.
 25( y  2)( y  2)
Check the answer by multiplying the factors. The check may be completed
mentally.
ALGEBRA CONCEPTS
a( b + c ) = ab + ac
( a + b ) ( a - b ) = a 2 - b2
( a + b ) 2 = a 2 + 2ab + b 2
( a - b ) 2 = a 2 - 2ab + b 2
;
a 3 + b 3 = ( a + b ) ( a 2 - ab + b 2 ) ;
a 3 - b 3 = ( a - b ) ( a 2 + ab + b 2 )
ab
a
b


c
c
c
a
a
a


bc
b
c
but
a
c
ad  bc


b
d
bd
Reduce factors only:
ax  ay
 x y
a
but
ax  y
 x y
a
FACTORING : More Examples
Ex.
2 x 2 - 50
2 ( x 2 - 25 )
2(x - 5)(x + 5)
Ex. x 2 - 6 x + 9
(x - 3 ) 2
Ex. 6 x 2 + 7x - 3
( 3x - 1 ) ( 2x + 3 )
Ex. 6 x 2 - 11 x + 3
( 3x - 1 ) ( 2x - 3 )
Ex. 8 x 3 + 27
(2x) 3 + (3) 3
( 2x + 3 ) ( 4x 2 - 6x + 9 )
Ex. 3x 3 + 6x 2 - x - 2
3x 2 ( x + 2 ) -1 ( x + 2 )
( x + 2 ) ( 3x 2 - 1 )