# Thread: The art of factoring....

1. ## The art of factoring....

I'm a calculus 3 student and the bane of my existence has ALWAYS been factoring. It's just something I constantly struggle with and never learned formally. I'm always BLINDLY taking guesses and "brute forcing" factors with my calculator (which can take well over 25-50 tries easily, and most of the time I never find the answer).

Is there any way of doing it systematically? Does anyone know a good method? (I'm a calc 3 student, so I don't own an algebra book)

I can look at any polynomial like

$\displaystyle f(x) = 12x^3-12x^2-24x$
which factors to
$\displaystyle 12x(x-2)(x+1)$

But I cannot see why! It looks like magic to me.

Ok... So I know simple factors like:

$\displaystyle x^2+3x - 4$

Which comes down to setting up
$\displaystyle (x+a) (x-b)$
And finding something that multiplies to 4 and adds to 3. Those are easy. But honestly, those are about the ONLY ones I can do, and I'm coming across harder ones in my book... Although, I rarely see any that go above a cubic polynomial

Any advice/help would be appreciated. Thanks.

2. Originally Posted by bobbooey
I'm a calculus 3 student and the bane of my existence has ALWAYS been factoring. It's just something I constantly struggle with and never learned formally. I'm always BLINDLY taking guesses and "brute forcing" factors with my calculator (which can take well over 25-50 tries easily, and most of the time I never find the answer).

Is there any way of doing it systematically? Does anyone know a good method? (I'm a calc 3 student, so I don't own an algebra book)

I can look at any polynomial like

$\displaystyle f(x) = 3x^4-4x^3-12x^2 + 5$
which factors to
$\displaystyle 12x(x-2)(x+1)$

But I cannot see why! It looks like magic to me.

Ok... So I know simple factors like:

$\displaystyle x^2+3x - 4$

Which comes down to setting up
$\displaystyle (x+a) (x-b)$
And finding something that multiplies to 4 and adds to 3. Those are easy. But honestly, those are about the ONLY ones I can do, and I'm coming across harder ones in my book... Although, I rarely see any that go above a cubic polynomial

Any advice/help would be appreciated. Thanks.
The factorization you have for $\displaystyle f(x)=3x^4-4x^3-12x^2+5$ is not correct!

Sometimes a good place to start is to guess the zero of the polynomial. For instance, if $\displaystyle x=-1$ then $\displaystyle f(-1)=3+4-12+5=0$. This tells us that $\displaystyle x+1$ is a factor.

Next, long divide (or use synthetic division) to reduce the degree of the polynomial. I leave it for you to show that

$\displaystyle \frac{3x^4-4x^3-12x^2+5}{x+1}=3x^3-7x^2-5x+5$.

Also observe that if $\displaystyle x=-1$, then $\displaystyle 3(-1)^3-7(-1)^2-5(-1)+5=-3-7+5+5=0$; therefore, $\displaystyle x+1$ is a factor of $\displaystyle 3x^3-7x^2-5x+5$. Apply long division again to get

$\displaystyle \frac{3x^3-7x^2-5x+5}{x+1}=3x^2-10x+5$.

Therefore, $\displaystyle 3x^4-4x^3-12x^2+5=(3x^2-10x+5)(x+1)^2$.

Observe that $\displaystyle 3x^2-10x+5$ cannot be factored nicely (it will involve square roots), so its best that we leave it like I did above.

Factoring in general is a game (to me anyways) and requires some intuition; there are a couple theorems out there that help make it a little easier (like the method I just used). Maybe you should take a look at those.

Does this make some sense?

EDIT: Take a look at this: http://tutorial.math.lamar.edu/Class...Factoring.aspx and see if this helps demystify the factoring process a little.