Question 51402
<pre>Can anyone give me any tips on factoring the greatest common monomial factor
from a polynomial. I dont have a problem to submit. I have a lesson sheet for
my class but the lesson isnt in the text book we use. If anyone can help I
appretiate it.
<pre><font size = 5><b>
6x³y - 9x³y²z + 12x²y³z²

We first look for a number we can take out. 
6, 9 and 12 can all three be divided evenly 
by 3, so we can take out a 3. So far,

GCF = 3....

Now we look at the letter x

6x³y - 9x³y²z + 12x²y³z²

The first term contains x³. The second term
contains x³ and the third term contains x².
Thus x appears in ALL factors so we can
take out the smallest power of x. The 
smallest power of x is x², so we can take 
out x².  So far

GCF = 3x²....

Now we look at the letter y

6x³y - 9x³y²z + 12x²y³z²

The first term contains y or y¹. The second
term contains y² and the third term contains y³.
Thus y appears in ALL factors so we can
take out the smallest power of y. The 
smallest power of y is y¹, so we can take 
out y¹, or just y.  So far

GCF = 3x²y....

Now we look at the letter z

6x³y - 9x³y²z + 12x²y³z²

The first term contains no z. Therefore we
cannot take out any power of z, because to
take out a letter, that letter must appear
in ALL the terms, so the final
GCF is

GCF = 3x²y

Now under the original expression

6x³y - 9x³y²z + 12x²y³z²

We write this GCF and a set of parentheses,
long enough to hold 3 terms.

3x²y(                )

We have three blanks to fill.

6x³y - 9x³y²z + 12x²y³z²
3x²y(__ - ____ + ____)

To fill the first blank we ask the question:
What must we multiply 3x²y by to get the 
first term 6x²y in the original?  The answer
is 2x, because the 3 needs to be multiplied by
2 and the x² needs to be multiplied by x, so
3x²y must by multiplied by 2x to get 6x²y 

6x³y - 9x³y²z + 12x²y³z²
3x²y(2x - ____ + _____)

To fill the second blank we ask the question:
What must we multiply 3x²y by to get the 
second term 9x³y²z in the original?  The 
answer is 3x¹y¹z or just 3xyz, because 
the 3 needs to be multiplied by 3 and the
x² needs to be multiplied by x¹, or just
x, and we must get the z, so we see that
3x²y must by multiplied by 3xyz to get 
9x³y²z.

6x³y - 9x³y²z + 12x²y³z²
3x²y(2x - 3xyz + ____)

To fill the third blank we ask the question:
What must we multiply 3x²y by to get the 
third term 12x²y³z² in the original?  The 
answer is 4y²z², because 
the 3 needs to be multiplied by 4. We do
not need any x's. The y needs to be 
multiplied by y², and we must get z²,
so we see that 3x²y must by multiplied by
4y²z² to get 12x²y³z²

6x³y - 9x³y²z + 12x²y³z²
3x²y(2x - 3xyz + 4y²z²)

That's the final result.

Edwin</pre>