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