Question 118367
To find either the Least Common Multiple ({{{LCM}}}) or Greatest Common Factor ({{{GCF}}}) of two numbers (or expressions), you always start out the same way: you {{{find }}}the {{{prime }}}{{{factorizations}}} of the two numbers(or expressions).


so, we will factor it first:


{{{x^3 + 10x^2 + 25x}}}

={{{x(x^2 + 10x + 25)}}}………notice that {{{ x^2 + 10x + 25}}} is square of the sum of two numbers

={{{x(x + 5)^2}}}………

{{{x(x + 5)(x + 5)}}}……… the {{{prime}}}{{{ factors}}}



Now factor the other expression:


{{{x^4 + 5x^3 }}}……..factor out common {{{x^3}}}

={{{x^3(x+ 5) }}}……..or…..{{{x*x*x(x+ 5) }}}…….. the {{{prime}}}{{{ factors}}}



Now write them in order:

{{{ x^3 + 10x^2 + 25x = x(x + 5)(x + 5)}}}………

{{{x^4 + 5x^3 ………..= x*x*x(x+ 5) }}}


since the {{{LCM}}} is the smallest expression that both {{{x^3 + 10x^2 + 25x}}} and {{{x^4 + 5x^3 }}} will divide into, or it is the smallest expression that contains both {{{x^3 + 10x^2 + 25x}}} and {{{x^4 + 5x^3 }}} (that both expressions fit in to), we will have:


{{{LCM = x*x*x(x + 5)(x + 5) = x^3(x^2 + 10x + 25)}}}………


since the {{{ GCF }}}is the biggest expression that will divide into both {{{x^3 + 10x^2 + 25x }}}and {{{x^4 + 5x^3 }}}(inother words, it's the expression that contains all the common factors), the {{{GCF}}} is the {{{product}}} of any and all factors that {{{x^3 + 10x^2 + 25x}}} and {{{x^4 + 5x^3 }}} share. It will be:

{{{GCF = x(x+5) = x^2 + 5x}}}