Finding The Greatest Common Factor. Greatest Common Factor: of two numbers is the result of two numbers being factored into their smaller factors individually and the of all the numbers that are factors, the one that is greatest is the greatest common factor:
Symbol: ( ) means greatest common factor. For instance, (6,8) means the greatest common factor of 6 and 8.
First, find all the factors of 6. The factors of 6 are all the numbers that when multiplied by another number give us 6. Those are, 1, 2, 3, 6. This is so because:
1*6=6 and 2*3=6 So you see that each number when multiplied by another number gives us 6.
Second, Find the factors of 8. Those are, 1, 2, 4, 8. So, now, when we look at the factors of 6 and 8
for 6: 1, 2, 3, 6; for 8: 1, 2, 4, 8. We see that 1, and 2, both appear in the factors of both six and 8: Now, of the factors that appear in both numbers, that is 1 and 2, the greatest one is 2. *
We use greatest instead of greater because when we speak of bigger numbers, there might be more than just two numbers that are common factors.
Factoring A Difference Of Two Squares
Once students are familiar with the idea of factoring, there are shortcuts to the FOIL method of quadratic equation factoring. One of these is factoring a difference of two squares. A difference of two squares means that we have a monomial multiplied to another monomial to give us a quadratic function where one monomial is a conjugate pair of the other and the middle term in the quadratic form of an equation disappears.
A quadratic equation is represented by ax^2+bx+c, when the "bx" term disappears, we have a difference of two squares. Example: (x-3)*(x+3), here (x+3) is a conjugate pair of (x-3), and here is where we get a difference of two squares. (x-3)(x+3), using the FOIL method, is:
(x)(x)+(x)(3)+(-3)(x)+(-3)(3)=
x^2+3x-3x-9
x^2-9.
Here, we see that since the middle term disappeared, we see that (x-3)(x+3) is simply X^2-(3)^2, the two squares being X and 3, which is why it is called the difference of two squares.
Of course, the same thing would result if we were doing (x+3)(x-3) since multiplication is commutative.
Now, to give more examples:
(x+4)(x-4)= (x^2-4^2)=(x^2-16)
(x+9)(x-9)=(x^2-9^2)=(x^2-81)
And so on, from here on, the patern is too clear and it would be too much repetition to go on.
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