Is there more than one way to factor this? Show your answer using both words and mathematical notation? Provide an expression for your classmate to factor.
This is actually a very good question to develop, if not evaluate, a student's understanding of factoring. Let's do a quick factoring lesson:
When factoring a trinomial (in general), the first thing to look for is a GCF (Greatest Common Factor) between each of the three terms in the trinomial. Think back to early middle school; the GCF of two numbers is a whole number that divides into both of the two numbers.
Example 1) The GCF of 20 and 15 is 5.
Example 2) The GCF of 24 and 18 is 6. Yes, 2 also divides into both 24 and 18, but we are looking for the LARGEST number that goes into both.
Once a GCF has been factored out (or if there is not a GCF), the next thing to look at are pairs of numbers that multiply to equal c/a (Remember: c/a is the product of the roots of a trinomial (You may not have learned this yet - tip for the future!). Also, if a=1, then you're just looking for pairs of numbers that multiply together to get c.).
Now compare the pairs of numbers - which pair of numbers add up to b/a? (Remember: -b/a is the sum of the roots of a trinomial, but the positive or negative sign of the roots is opposite the number that is actually in the parenthesis when you factor. For example: (x-4)(x+2)=0 --> The roots for this equation are x=4 and x=-2, but notice how the numbers in the parenthesis are actually -4 and +2. Also, if a = 1, you're just looking for a pair of numbers that add together to get b.)
Last step is to put the numbers in the correct position in the GCF(_x ± #)(_x ± #) basic format for factoring trinomials.
So, to recap:
1. Look for GCF.
2. Identify pairs of numbers that multiply to get c/a.
3. Which pairs of numbers in Step #2 add to get b/a?
4. Plug the numbers & GCF into the right spot in the factoring format.