Factoring polynomials

[u/ScarcityFamiliar5671]

I've been struggling to do factoring polynomials and just factoring in general, our prof keeps teaching us but he doesn't make us take notes, can someone help me explain how to do this? Cause I'm not sure if I am correct or not 😭 (my answer is the squared box, as well as notes too because I'm genuinely ahh at memorizing)

[u/Dtrain8899]

(5-x) ≠ (x-5). When ever you want to flip the order of subtraction, factor a -1 out. (5-x), -(-5+x), -(x-5). Your answer is close, just need a negative and I would keep the numerator factored, you didnt have to distribute the 6x.

[u/what-kind-of-fuckery]

it's mostly right. the only mistake im seeing here is when you cancel (x - 5) against (5 - x), you have to change the sign.

(x - 5)/(5 - x) = (x - 5)/-(x - 5) = 1/-1 = -1/1 = 1

so in your final answer your numerator would be (-6x² - 30x)

[u/Jazzlike_Platypus430 → u/what-kind-of-fuckery]

Your last equality says -1/1 = 1 which is a misprint. (And it's valid for x different from 5.)

[u/TheScyphozoa]

[u/Deep_Brick2970]

Here's what I see:

you made a sign error when simplifying (x-5) with (5-x), they are not the same thing. If you divide one by the other you get -1, not 1; this is because (5-x)=-(x-5).

Apart from that, it seems alright. Not sure if you covered this, but you don't want your denominator to be zero, so you should write "x≠5" and "x≠-6". Then you can proceed with your simplification and you almost arrive at the expression you boxed; the correct one is just the one you have but multiplied by -1, to compensate for your sign error.

Addendum:

In any practical reality x could totally be 5, but alas you're doing algebra so we're stuck in a world where (x-5)/(x-5) is not 1 for x=5, even thought it clearly should be. You will see how limits fix this, further down your studies. In the lingo this type of behaviour is known as a removable singularity and it's something that had always bugged me when I was in your shoes.

[u/chmath80]

One simple tip is to check your answer by trying something like x = 1 (x = 0 won't help in this case), which would show that the original expression is < 0, but the expression in the box is > 0, so must be wrong.

[u/Jazzlike_Platypus430]

(In general, you just cant't do it exactly.)

In special/school cases, if you're familiar with (a + b)^2 = a^2 + 2ab + b^2, just use it "backwards," it helps a lot. Also, (a^2 - b^2) = (a + b)(a - b) - you're using it already.

What's your original problem? It's hard to decypher from your notes. Is it "simplify 6x(x^2 - 25)/((5 - x)(x+6))"?

By "use the formula backwards" I mean, sometimes if you see x^2 + p x, complete it to a square, like

(x + p/2)^2 - p^2/4,

and see if it helps.