1 This Factoring Packet was made possible by a GRCC Faculty Excellence grant by Neesha Patel and Adrienne Palmer. FACTORING HANDOUT A General Factoring Strategy It is important to be able to recognize the different types of polynomials and know each ones factoring method. Use these steps to guide you: 1) Factor out the greatest common factor (GCF),if there is one. 2) Are there two terms? (Binomial) Is it Difference of two squares? If yes, factor by using: ( )( ) Page 2 Note: You cannot factor a binomial in the form . 3) Are there three terms? (Trinomial) Is it a perfect square trinomial? If yes use: ( ) Page 3 ( ) 4) Is the form where ? Factor by Product-Sum Method . (Page 5) 5) Is the form where ? Factor by Guess and Check, the ac-Grouping or one of the other methods attached. (Page 6 and 7) 6) If you can’t factor it by any method above, the polynomial is irreducible. It is prime.
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This Factoring Packet was made possible by a GRCC Faculty Excellence grant by Neesha Patel and Adrienne
Palmer.
FACTORING HANDOUT
A General Factoring Strategy
It is important to be able to recognize the different types of polynomials and know each ones factoring method.
Use these steps to guide you:
1) Factor out the greatest common factor (GCF),if there is one.
2) Are there two terms? (Binomial)
Is it Difference of two squares? If yes, factor by using:
( )( ) Page 2
Note: You cannot factor a binomial in the form .
3) Are there three terms? (Trinomial)
Is it a perfect square trinomial? If yes use:
( ) Page 3
( )
4) Is the form where ? Factor by Product-Sum Method . (Page 5)
5) Is the form where ? Factor by Guess and Check, the ac-Grouping
or one of the other methods attached. (Page 6 and 7)
6) If you can’t factor it by any method above, the polynomial is irreducible. It is prime.
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Factoring Binomials
Difference of Two Squares: ( )( )
Example: Factor
*Notice that both and 9 are perfect squares: and
So ( )( )
Example: Factor
Factor out the GCF first: ( )
*Notice that both and are perfect squares: ( ) and
*Notice that both and 9 are perfect squares. So a good first guess at how to
factor this trinomial would be to use their roots:
( )( )
Then we can just work on figuring out what signs need to go in each parentheses.
With a little trial and error, we see that a minus sign in each parentheses would
work.
( )( )
So ( )( )
( )
Another way that we could have looked at this factoring problem would be to
notice that and and ( )( ) [if we are trying to match
things up with the special factoring patterns for perfect square trinomials, then
( )( ) ].
Recognizing the special factoring pattern ( ) , we could
have factored immediately into the form ( )
Example: Factor
*Notice that both and 25 are perfect squares. So a good first guess at how to
factor this trinomial would be to use their roots:
( )( )
Then we can just work on figuring out what signs need to go in each parentheses.
With a little trial and error, we see that a plus sign in each parentheses would
work.
( )( )
So ( )( )
( )
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Another way that we could have looked at this factoring problem would be to
notice that and and ( )( ) [if we are trying to
match things up with the special factoring patterns for perfect square trinomials,
then ( )( ) ].
Recognizing the special factoring pattern ( ) , we could
have factored immediately into the form ( )
Example: Factor
*With a little practice, you may notice that ( )( )
So using the special factoring pattern ( ) we get
( )
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FACTORING TRINOMIALS OF THE FORM cbxax 2 ( )
USING THE PRODUCT-SUM METHOD
(Use when a=1)
Example: 652 xx
STEPS 1. Setup the binomial factors and enter the first term of each factor. Remember, you’re
doing the reverse of FOILING.
(x )(x )
2. Write the value of “b” and “c”: b = 5 , c = 6
3. List all pairs of integers whose product is c.
C= 6
32
61
4. Choose the pair whose sum is b:
b = 5
(This one)
5. Plug the matching pair into the binomial factors:
)3)(2( xx
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FACTORING TRINOMIALS IN THE FORM
FACTOR BY GUESS AND CHECK
Use this method if the a and c values are small or prime.
E.g. Factor
Think reverse FOILing. The only choice for the first terms in each binomial is 5x and x to obtain
the product of that appears in the first term above.
( )( )
We wish to obtain the c value of 2 when we FOIL back. Our factors of 2 are 1 and 2. So either we
have:
( )( ) Or
( )( )
FOILing the first option gives the middle term of that appears in the original trinomial.
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FACTORING TRINOMIALS IN THE FORM
FACTOR BY THE “ac” AND GROUPING METHOD
Use when or 0:
Example 1:
STEPS: 1. Factor out a GCF if there is one. This example does not have one.
2. Then use the steps below to factor the trinomial into two binomial factors.
3. List the values of a, b and c in the expression:
4. Find the product of “ ”:
5. List the factor pairs that give the product of :
6. Find the pair of factors whose sum equals “b”, and write as (i.e. The middle term including the variable)
7. Replace these two terms for bx in the original expression, so that you now have an expression with 4 terms:
8. Use the Grouping Method to complete the factoring as follows: Group the first two terms together and the last two terms together:
( ) ( ) 9. Factor out any common factors from the first group and any from the second group:
( ) ( )
Ist term 2nd term
Notice that we now have an expression with just 2 terms. Each term should have a common factor (2x + 1 in this case).
10. Factor out this common factor from each term: ( )( ). These are your binomial factors.
11. FOIL out to double check that your factors match the original equation.
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FACTOR BY THE “ac” AND GROUPING METHOD continued:
Example 2:
1. Factor out the GCF: ( )
2. Ignore the GCF for now. We wish to factor . List the values of a, b and c for this quadratic expression:
a = 2, b = -15, c = -27
3. Find the product of “ ca ”: ( )
4. List all factor pairs of the product in step 3. Be systematic and keep going until you find
the pair whose sum equals (the value of b). Notice that the signs of the factors must
be opposites:
( )( ) ( )( ) for example. The two factors we want are 3 and
Rewrite the middle term using these 2 factors:
5. Replace these two terms for bx in the original expression, so that you now have an expression with 4 terms:
6. Group the first two terms together and the last two terms together: ( ) ( )
7. Factor the GCF from each set of parentheses: ( ) ( ) Notice that we now have an expression with just two terms.
Each term has a common factor of . 1st term 2nd term
8. Factor out this common factor from each term to obtain your two binomial factors: ( )( )
Note: If you had reversed the two middle terms in step 5 to obtain Be careful how you handle the parentheses. If you have a minus outside the second set of parentheses, you will need to change the sign of every term inside the parentheses as follows:
( ) ( ) Both the 18x and 27 change signs.
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FACTORING TRINOMIALS IN THE FORM cbxax 2
USING THE TABLE METHOD
Use when 1a or 0:
STEPS:
1. Example: 372 2 xx
2. Factor out a GCF if there is one. Then use the steps below to factor the remaining trinomial.
3. List the values of a, b and c in the quadratic expression: a = 2, b = 7, c = 3
Setup a box as shown below. Write the value of in the top left unshaded box and c bottom right
unshaded box.
3. Above example
5. Find the product of “ac”:
632
6. List factors of the product in step 5: 61 , 23
7. Find the pair of factors whose sum equals “b”, and write as
xx 61 (i.e. The middle term, i.e. include the variable)
8. Plug these two terms into the two unshaded empty boxes in the table. (it doesn’t matter
which term goes into which box). Then factor out the common factors in each row and column and place
these in the shaded boxes:
Factor
First term 2ax
Last term c
Factor
22x
3
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9. The two shaded boxes give you the factored binomials products:
)3)(12( xx
FOIL out to check you get the original trinomial expression.
Factor x 3
2x 22x 6x
1 x 3
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Practice Problems
Begin by doing all of the problems that end with a 7 (problems 7, 17, 27, etc.). Check your answers by
multiplication; if you multiply your answer out and simplify, you should get the original polynomial. For problems
that you have trouble with, work on the other nine problems in that group. Again, check your answers by