SOLUTION: How do I find the LCM and GCF of {{{x^3+10x^2+25x}}} and {{{x^4+5x^3}}}?

How do I find the LCM and GCF of
 and ?

You factor them all.



First take out an x



Factor the trinomial in the parentheses.

First write this:

x(x    )(x    )

Think of two whole numbers which have product
+25 and sum +10.  They are +5 and +5. So the
factorization is



or



Now factor



by taking out 



Now we look at all the factors of both

 and 

It may help to write out the factors without exponents:

 and 

To find the LCM. 

Notice that the factor  occurs
1 time in the first factor and 3 times in the second.

1 is the LEAST, so the GREATEST COMMON FACTOR contains
x the LEAST number of times, which is 1 time.  

3 is the GREATEST, so the LEAST COMMON FACTOR contains 
x the GREATEST number of times, which is 3 times.

So far write:  

GCF = x            LCM = x³

Notice that the factor  occurs
2 times in the first factor and 1 time in the second.

1 is the LEAST, so the GREATEST COMMON FACTOR contains (x-5) 
the LEAST number of times, which is 1 time.  

2 is the GREATEST, so the LEAST COMMON FACTOR contains (x-5) 
the GREATEST number of times, which is 2 times.   

So we end up with the complete GCF and LCM:  

GCF = x(x+5)       LCM = x³(x+5)

Remember: 

1. To get the GREATEST common factor you use each prime factor
the LEAST number of times that it appears in any expression.

2. To get the LEAST common multiple you use each prime factor 
the GREATEST number of times that it appears in any expression.

Edwin