Here's one I came across today:
Factor by grouping.
x^3+3x^2+15x+125

Question

Here's one I came across today:
Factor by grouping.
x^3+3x^2+15x+125

matheducatormcg · Answer

$$x^3+3x^2+15x+125$$

anonymous · Answer

(x^3 + 3x^2) + (15x + 125)

Try factoring each one of those separately.

matheducatormcg · Answer

won't work.  you have to use commutative property to get proper grouping.

anonymous · Answer

Actually, hold on...let me try and solve this one first.

matheducatormcg · Answer

try not to use a calculator

matheducatormcg · Answer

my summer school students thought the question was a little unfair how the terms were ordered in question.

matheducatormcg · Answer

well, it is factorable.

anonymous · Answer

Ah. Got it.
$$x^3 + 125 + 3x^2 + 15x$$
$$(x+5)(x+5)(x+5) + 3x(x+5)$$
$$(x+5)((x+5)^2 + 3x)$$
$$(x+5)(x^2 + 13x + 25)$$

anonymous · Answer

$$3(x+5)^2(x^2+5x+25)$$ I think that looks right

anonymous · Answer

Ah, perhaps I forgot to factor out the x from the (x+5)... it's late.

matheducatormcg · Answer

you got the cubic factoring wrong.
here is solution:

$$x^3+3x^2+15x+125$$
$$=(x^3+125)+(3x^2+15x)$$
$$=(x+5)(x^2+5x+25)+3x(x+5)$$
$$=(x+5)((x^2+5x+25)+3x)$$
$$=(x+5)(x^2+8x+25)$$

pretty cool.  for a factoring.

anonymous · Answer

Shouldn't that be x^2 + 10x + 25?

matheducatormcg · Answer

No.  5x +3x is 8x.  ??

the cubic is a lurker, thinking that 
$$(x^3+125)=(x+5)(x+5)(x+5)$$
factors this way is common mistake

anonymous · Answer

Oh. You're right.  That's my fault for not putting too much thought into it, heh.

matheducatormcg · Answer

thanks for checking question out.