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