what is prime factorization?
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What is prime factorization?. Maybe use this number as an example? -117. So final answer is: -1 x 3 2 x 13. -1 3 39. 3 13. GCF – Greatest Common Factor. Find the GCF of each set of monomials. 54, 63, 180. 9. 27a 2 b & 15ab 2 c. 3ab. 8g 2 h 2 , 20gh, 36g 2 h 3. - PowerPoint PPT PresentationTRANSCRIPT
What is prime factorization?
Maybe use this number as an example?
-117
-1 3 39
3 13
So final answer is:
-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
55p2 – 11p4 + 44p5
-1 2 2 2 2 3 c d d
2 2 5 7 x x x y y z
11p2(5 – p2 + 4p3)
Factor completely:
12ax + 3xz + 4ay + yz
(3x + y) (4a + z)
Since all terms do not have a common factor, use grouping:
(12ax + 3xz) + (4ay + yz)
3x (4a + z) + y (4a + z)
Factoring Trinomials
ax2 + bx + cRemember 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)
Always follow
these steps!
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?2. Is there a GCF?YES … so factor if out
3. Is it a special pattern?4. Can it be factored any further?
You’re done!
NO!
NO!
Another Example
4x2 – 16 4(x2 – 4)
1) Can it be simplified?2. Is there a GCF?YES … so factor if out3. Is it a special
pattern?4. Can it be factored any further?Ta da … you’re done!
YES – it’s the difference of squares
so 4(x + 2)(x – 2)
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?
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?
You’re done!
Trinomial Examples #2
x2 + 3x – 10 (x + 5)(x – 2)1) Can it be simplified?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?
You’re done!
Trinomial Examples #3
2x2 – 11x + 15 (2x – 5)(x – 3)
1) Can it be simplified?2. Is there a GCF?
3. Is it a special pattern?4. Factor … use the method of
YOUR choice!
You’re done!CAREFUL – there’s a number in front of
the x2!I’ll wait while you work it out …..
Trinomial Examples #4
4x2 – 18x – 10 2(2x2 – 9x – 5)1) Can it be simplified?2. Is there a GCF?
3. Is it a special pattern?
4. Factor … use the technique of YOUR choice!
You’re done!
CAREFUL – there’s a number in front of
the x2!I’ll wait while you work it out ….. 2(x – 5)(2x + 1)
Difference of Squares
a2 – b2 (a + b)(a – b)
Example:
4x2 – 25 (2x + 5)(2x – 5)
2x 2x 5 5
What would you do?
48a2b2 – 12ab
6x2y – 21y2w +24xw
xy – 2xz + 5y – 10z
What would you do?
a2 – 10a + 21
3n2 – 11n + 6
9x2 – 25
x2 – 6x – 27 = 0