~Factoring~

~Every once in a while in limit expressions it is necessary to factor them in order to get the correct answer. Usually, they will occur in limits that result in 0/0 when substituting into the expression.

Factoring: Factoring an number or expression means to express it as a product of numbers or product of expressions. Prime factoring of numbers will express the number as a product of the prime numbers that make it up. For example, say you would like to factor the number 105 into primes. Start with the smallest prime (2) and see it it divides it. It does not. Then try 3 (the next prime). It does and gives 35 as a quotient. So far, we have 105=(3)(35). It is factored but not into primes. Doing the same with the quotient 35, we will get 35=(5)(7). So, combining the results, we have 105=(3)(5)(7). It is now factored into primes.

In algebra, the most common type of factoring is taking out a common factor from each term. This is just the opposite of the distributive law in mathematics. For example, take the two term expression xb-yb. We can see that b is a common factor in each term. So, take it out front and express it this way: b(x-y). Sometimes there are more expressions that are common to each term. For example, 3(x²)(y³) + 12xy will have 3xy as the greatest commom factor in both terms. So, take that out front and write it in factored form, 3xy(xy² + 4). You can always check to see if you factored it correctly by multiplying it out to see if you get the original expression back.

Difference of two squares: Expression like x² - y² can be factored into (x-y)(x+y). Some times the squares are more complicated like, (a²)(b⁴) -(a⁴)(b²).
In this case, we get take out the common factor of (a²)(b²) and write it as [(a²)(b²](b² - a²), then finally writing it as [(a²)(b²](b-a)(b+a).

~Factoring the Sum and Difference of Cubes~

~Both forms are factored in very similar ways. There are just sign differences.

~The Sum of two Cubes~

a³ +b³ = (a+b)(a²-ab+b²)

~The Difference of two Cubes~

a³ -b³ = (a-b)(a²+ab+b²)

Also, see my article on my website dealing with expanding binomials. This will be very helpful in computing the derivatives, using the limit definition, for expressions containing x to powers of 3 or higher. Look for the Binomial Expansion link.