4x^9 + 8x^8 - 12x^7 + 8x^6
If you have trouble finding GCF's, it can be helpful to factor each term "fully". By "fully" I mean:- Factor each coefficient into prime numbers
- Rewrite the variables without exponents
Let's see:
4x^9 = 2 * 2 * x * x * x * x * x * x * x * x * x
8x^8 = 2 * 2 * 2 * x * x * x * x * x * x * x * x
12x^7 = 2 * 2 * 3 * x * x * x * x * x * x * x
8x^6 = 2 * 2 * 2 * x * x * x * x * x * x
Note how I used spacing the ensure that each column has the same factor in it. Lining up the factors like this can also be helpful in finding the GCF. In the table above a factor that is common to all four terms will be the factor in any column that has four entries. The GCF will be the product of all the common factors. Looking at the table we can see that there are 2 columns of 2's and 6 columns of x's that have an entry in all four rows. So the GCF is:
GCF = 2 * 2 * x * x * x * x * x * x = 4x^6
P.S. There are other uses for factoring and lining up the factors like this:- Figuring out "what's left" of each term if the GCF is factored out. Just look at the GCF line of factors and compare it to a term's line. The factors that are in the term's line but not in the GCF's line of factors will be "what's left" of the term after the GCF is factored out. For example:
GCF = 2 * 2 * x * x * x * x * x * x
8x^6 = 2 * 2 * 2 * x * x * x * x * x * x
From this we can see that the only factor in the term's line that is not in the GCF line is that 3rd 2. So if the GCF is factored out of 8x^6 there will be a 2 left. Or
GCF = 2 * 2 * x * x * x * x * x * x
12x^7 = 2 * 2 * 3 * x * x * x * x * x * x * x
Here we can see that the term has factors of a 3 and a 7th x that are not in the GCF list. So if we factor out the GCF from this term we will have 3 * x or just 3x left.
Note: If a term is the same as the GCF then there will be just a 1 (one) left if the GCF is factored out of this term.
- Finding least common multiples (LCM's). The LCM will be the product of all the different columns. For our four terms above the LCM will be:
LCM = 2 * 2 * 2 * 3 * x * x * x * x * x * x * x * x * x = 24x^9
Note: Lowest common denominators (LCD's) are simply the LCM of some denominators. So we can use this method to find LCD's, too. Not only that we can figure out each fraction need to be multiplied by to turn the denominator into the LCD. For example, let's say we wanted to add
We would need the LCD. Since the denominators are the same as the terms we've already factored we can use the LCM we found above:
LCM = 2 * 2 * 2 * 3 * x * x * x * x * x * x * x * x * x = 12x^9
Then if we compare the factors of the LCM with the factors of each denominator we can see what factors are "missing". For the first denominator:
4x^9 = 2 * 2 * x * x * x * x * x * x * x * x * x
We can see that this term's factors are "missing" the third 2 and a 3. So we would multiply the numerator and denominator of the first fraction by 2*3 or 6. For the third denominator:
12x^7 = 2 * 2 * 3 * x * x * x * x * x * x * x
We can see that this term's factors are missing the third 2 and the 8th and 9th x's. So we would multiply the third fraction's numerator and denominator by 2 * x * x or
Note: If a denominator is the same as the LCD then it does not need to change. Don't multiply it by anything.