The Greatest Common Factor – (G
The Greatest Common Factor – (G.C.F.)
If you remember elementary school, your teacher introduced the G.C.F.
Let’s say you have three numbers 18 54 81
Notice, they all share (3)(3) in common, therefore the G.C.F. is 9.
This is the same when we work with polynomials.
Eg [pic]
REMEMBER: When the question says FACTOR, look for the G.C.F. FIRST!!!
Factoring out a Binomial
Eg [pic]
Eg [pic]
Factoring by Grouping
Grouping 2 & 2
Eg
ab + ac + db + dc [pic]Factor by grouping the question 2 and 2.
Factor out the common binomial (b+c)
Some questions will make you re-arrange the terms to be able to group them.
Grouping 3 & 1
Eg
[pic] change to: [pic]
Factor the trinomial inside the brackets [pic]
You now have a difference of two squares [pic]
Substitute a variable for the binomial. Let “p” =(x - 2)
So now you have [pic]
Factor: [pic] Substitute (x – 2) back in for “p”.
[pic] You’re done!
Factoring in the form of “ax2 + bx + c” when ‘a’ = 1 Look for GCF
Eg
[pic] ( Look For G.C.F.
← See how many terms are in the question
( If there are 3 terms it is probably a regular quadratic or a
perfect square trinomial. (Geek Speak for x2)
This example has three terms so you are looking for 2 numbers that multiply to equal15 and those same two numbers add to equal 8.
X = 15 Remember ‘a’ = 1 and 1x15 =15
+ = 8 3x + 8x = 11x
Therefore: [pic] To check if you are correct, FOIL!
Factoring in the form of “ax2 + bx + c” when ‘a’ ( 1 Look for GCF
Eg
[pic] ( Look for G.C.F. (2)
( See how many terms. (3)
[pic] ( Find two numbers that multiply to equal 40.
Remember ‘a’ = 5 and 5 x 8 = 40
Those same two numbers must add to equal (– 14)
Method 1: Decomposition Look for GCF
[pic]
Method 2: AC Method Look for GCF
[pic]
**If the fractions cannot be reduced, then place the denominators (numbers on the bottom) in front of the variables (letters).
Special Quadratics Look for GCF
Perfect Square Trinomials (P.S.T.) [pic]
← If you cannot recognize the quadratic as a “perfect square trinomial”, don’t panic.
( You can factor the trinomial using either the “decomposition” or “the Taiwanese”
method.
← If the 1st and 3rd terms are perfect squares, then you may have a (P.S.T.).
( If the square root of the 1st term and the square root of the 3rd are multiplied
together and then doubled, and the result is the middle term, then you have a
(P.S.T).
Eg [pic]
Eg [pic] Look for two numbers that multiply to give you 36
and add to give you 12.
[pic] Remember to place the 4 back under the second term.
[pic]
[pic]Now place the denominator in front of the
first term.
A Difference of Two Squares [pic] Look for GCF
← The first thing to notice is that there are only two terms. They are both perfect squares.
( Secondly, they are separated by a ( [pic] ) sign. NOT a (+ ) sign.
← Put the square root of each term in each bracket. Then put a (+) sign in the middle of one bracket and a ( [pic] ) sign in the other bracket.
Eg [pic]
Eg [pic]**Notice the second bracket is a difference of two
squares. NOT THE FIRST BRACKET!!!
[pic] This is completely factored.
Note: The sum of two squares cannot be factored!!! Eg [pic]
Factoring by Substitution Look for GCF
Eg [pic] Substitute a single variable for (2x+3y). Let ‘w’ = 2x+3y
[pic] Now it is a difference of two squares question.
[pic]Substitute 2x + 3y back in for ‘w’.
[pic] Done!
Eg [pic] Let ‘y’ = (p+q)
[pic] Find two numbers to multiply to equal 15
and add to equal 8.
[pic] Substitute p+q back in for ‘y’.
[pic] C’est fini!
Eg [pic] Let (2 – x) = h
[pic] [pic] Now substitute (2 - x) back in.
[pic] The inside brackets
are necessary to prevent sign errors.
REMEMBER: When the question says FACTOR, look for the G.C.F. FIRST!!!
-----------------------
(3)(3)(3)(2)
(2)(3)(3)
(3)(3)(3)(3)
+ d(b+c)
a(b+c)
(a+d)(b+c)
5
3
5
3
-
4
- 10
40
=
x
- 4
- 10
- 14
=
+
Remember to reduce fractions to lowest terms.
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