What is prime factorization?
Maybe use this number as an
example?
-117
So final
answer is:
-1
3
39
3
13
-1 x 32 x 13
GCF – Greatest Common Factor
Find the GCF of each set of monomials.
54, 63, 180
9
27a2b & 15ab2c
 3ab
8g2h2, 20gh, 36g2h3
 4gh
Relatively Prime
• Define relatively prime, then give an
example.
If two or more integers or
monomials have a GCF of 1,
then they are said to be
relatively prime.
Example: 21m and 25b
Factor completely:
• 140x3 y2 z
-48cd2

2257xxxyyz
 -1 2 2 2 2 3 c d d
55p2 – 11p4 + 44p5
 11p2(5 – p2 + 4p3)
Factor completely:
12ax + 3xz + 4ay + yz
Since all terms do not have a common factor, use grouping:
(12ax + 3xz) + (4ay + yz)
3x (4a + z) + y (4a + z)
 (3x + y) (4a + z)
Factoring Trinomials
2
ax
+ bx + c
Remember to do and check each step:
1) Can the equation be simplified?
2) Is there a GCF? (then take it (factor it) out!)
3) Is it a special pattern: a2 – b2, a2 – 2ab + b2, a2 + 2ab + b2
look for perfect squares!!!
4) No special pattern, then factor! (Use grouping, ac method,
illegal or diamond factoring if necessary)
a2 – b2 = (a + b)(a – b)
a2 – 2ab + b2 = (a – b)2
a2 + 2ab + b2 = (a + b)2
Examples
4x2 + 16
 4(x2 + 4)
1) Can it be simplified?
NO!
2. Is there a GCF? YES … so factor if out
You’re
3. Is it a special pattern?
NO!
4. Can it be factored any further?
done!
Another Example
4x2 – 16
 4(x2 – 4)
3.if out
YES
Is–+itit’s
a special
the difference

4(x
2)(x
– pattern?
2) of squares
2.
Is there
…
factor
GCF?
1)YES
Can
it so
besoasimplified?
4. Can it be factored any further?
Ta da … you’re done!
Did you notice the similarity and the
differences between the last 2 problems?
Trinomial Examples
x2 + 7x + 12
 (x + 4)(x + 3)
1) Can it be simplified?
You’re done!
2. Is there a GCF?
3. Is it a special pattern?
4. Factor … what are the factors of the last
term that add up to the middle term?
Trinomial Examples #2
x2 + 3x – 10
 (x + 5)(x – 2)
1) Can it be simplified?
2. Is there a GCF?
You’re done!
3. Is it a special pattern?
4. Factor … what are the factors of the last
term that add up to the middle term?
Trinomial Examples #3
2x2 – 11x + 15  (2x – 5)(x – 3)
CAREFUL – there’s a number in
1) Can it be simplified?
2!
You’re
done!
front
of
the
x
2. Is there a GCF?
I’ll wait while you work it out …..
3. Is it a special pattern?
4. Factor … use the method of YOUR choice!
Trinomial Examples #4
4x2 – 18x – 10  2(2x2 – 9x – 5)
1) Can it be–simplified?
CAREFUL
there’s a number in
2. Is there
fronta GCF?
of the x2!
3. Is it a special pattern?  2(x – 5)(2x + 1)
I’ll wait while you work it out …..
4. Factor … use the technique of YOUR choice!
You’re done!
Difference of Squares
a2 – b2  (a + b)(a – b)
Example:
4x2 – 25
2x
2x 5
5
 (2x + 5)(2x – 5)
What would you do?
2
2
48a b
2
6x y
–
– 12ab
2
21y w
+24xw
xy – 2xz + 5y – 10z
What would you do?
– 10a + 21
2
3n – 11n + 6
2
9x – 25
2
x – 6x – 27 = 0
2
a