GCF, LCM, and Factoring by Gail

--> Hello Class!

In the last two days of class we have mostly covered GCF, LCM and Factoring.

GCF-This means the greatest common factor of the two or more numbers that you are comparing.

LCM-This is the least common multiple of the two or more numbers that you are comparing

We started the class off with finding the GCF and LCM of the following three numbers:

225p^3qr^2 135p^2q^2r^2 80p^3r^3

We first start with doing a factor tree for these numbers to find common factors, of the three numbers.

We will first start with the coefficients

Next we will create a venn diagram, showing what the common factors are between all three numbers. The center of the diagram, where all three numbers overlap is where you will find the GCF, by multiplying the middle numbers together. By multiplying all of the numbers on the venn diagram (counted once), equals the LCM. In this venn diagram, we will add in the variables.

Looking at this graph, and what I stated before, we can come to the conclusion that:

GCF- 45p^2r^2

LCM-2,700q^2r^3p^3

Next, we began to talk about factoring, there are three main types of factoring that we learned:

Here are the formulas that you plug the equations numbers into.

Trinomial Squares: a^2+2ab+b^2=(a+b)^2

a^2-2ab+b^2=(a-b)^2

Difference of Squares: a^2-b^2=(a-b)(a+b)

Sum and Difference of Cubes: a^3+b^3=(a+b)(a^2-ab+b^2)

a^3-b^3=(a-b)(a^2+ab+b^2)

Here are some examples of the three types of factoring, these are the examples that we did in class.

16a^2b+24ab^2=

8ab(2a+3b)

Steps:

Look for the GCF, then pull out the GCF putting it in front of the numbers that are left over after pulling out the GCF.

Trinomial Square:

32x^2-48xy+18y^2=

2(16x^2-24xy+9y^2) <---This matches the trinomial squared sequence (plug in numbers to formula)=

2((4x)^2-24xy+(3y)^2)=

2(4x-3y)^2

Trinomial Square:

x^6-2x^3y^2+y^4=

(x^3-y^2)^2

Plug in the numbers from the equation into the variables in the formula ex: a=x^3//b=y^2)

Difference of Squares:

81x^2-16y^2=

(9x-4y)(9x+4y)

We also talked about another type of factoring, which is factoring by grouping. You usually try this method when there are more than three terms.

Factoring by Grouping:

Is a way to group terms of an equation to factor out the common factor.

Example:

ab+3a-2b-6

The two groups are ab+3a and -2b-6

here in these two groups you pull out the common factor-which is (b+3)

a(b+3)-2(b+3)

Since the two common factors are the same, you can pull it out and multiply it by the left over numbers.

(b+3)(a-2)

In an equation with four terms you can also group three terms together, and leave one out.

Example-

x^2-4y^2-4x-4=

x^2-4x+4-4y^2<--In pink is a trinomial squared

(x-2)^2-4y<--Difference of two Squares

(x-2-2y)(x-2+2y)

Hope That Helps And Re-Caps Our Past Two Classes!!

:)

Examples and further explanation for these types of problems can be found on pages: 168-175