   SEARCH HOME Math Central Quandaries & Queries  Question from Ema, a student: Question 1: Factor completely. 3x(x - 4) + 5(x - 4) Question 2: Factor completely. x^2 + 2x + 3 Hi Ema,

Your first question involves a common factor. Let's look at a similar question.

Factor completely 3x2 - 5x

What you need to recognize is the x in both terms, in fact x2 = x x so you might see this expression as

3 x x - 5x

You can take one x out as a common factor so

3x2 - 5x = (3x - 5) x

You can't factor this any further so this is the expression in a completely factored form.

Common factors can be more complex than in my first example. Let's try

Factor completely 2x2(x + 2)- 4(x + 2)

Here I see (x + 2) as a common factor so

2x2(x + 2)- 4(x + 2) = (2x2 - 4)(x + 2)

but I'm not done since 2x2 - 4 has a common factor of 2 so

2x2(x + 2)- 4(x + 2) = (2x2 - 4)(x + 2) = 2(x2 - 2)(x + 2)

Now I'm done.

For your second problem x2 + 2x + 3 if it factors the factorization will look like

x2 + 2x + 3 = (x + a)(x + b)

where a and b are some numbers. If you expand the right side you get x2, two terms that contain x and ab. Hence, if my factoring was successful ab = 3. Thus a and b be 1 and 3 or and and b must be -1 and -3. Hence the factored form would have to be

(x + 1)(x + 3) or (x - 1)(x - 3)

Expand both of these. Do you get x2 + 2x + 3 in either case? If so then you have a factorization, if not then x2 + 2x + 3 doesn't factor.

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.