x^3+125; (x+5)
find the other factor

Question

x^3+125; (x+5)
find the other factor

whpalmer4 · Answer

We can write any polynomial as a product of factors.  That means 

$$x^3+125 = (x+5)(\text{other factors})$$Do you know how to divide polynomials, either by long division or synthetic division?

anonymous · Answer

no

anonymous · Answer

isn't there another way to solve it without division?

whpalmer4 · Answer

Well, it works pretty much the same way it does for numbers.  What's the biggest power of $x$ we can multiply with $(x+5)$ and still be able to subtract it from $x^3+125$?

anonymous · Answer

2?

whpalmer4 · Answer

We've got $x$, and we want to turn it into $x^3$, so we multiply by $x^2$:
$$x^2(x+5) = x^3 + 5x^2$$
Now we subtract that from our original polynomial:

$$x^3+125 - (x^3+5x^2) = -5x^2 + 125$$So our remainder after 1 division step is $$-5x^2+125$$  What's the biggest thing we can multiply with $(x+5)$ for the next division step?  We want to get a $-5x^2$ term...

anonymous · Answer

the x^3 cancels out right?

whpalmer4 · Answer

We've already taken out the x^3 term, yes.  now we are trying to divide $-5x^2+125$ by $x+5$

anonymous · Answer

how did u get -5x^2?

whpalmer4 · Answer

Here, maybe drawing it in the usual fashion will help:

             x^2  
     -----------------------------
x+5  | x^3  + 0x^2 + 0x  + 125
           x^3  + 5x^2
         --------------------
          0x^3 - 5x^2 + 0x + 125

anonymous · Answer

are you doing synthetic substitution?

whpalmer4 · Answer

now we're looking to subtract off the $-5x^2$ term, so we multiply $(-5x)(x+5) = -5x^2-25x$ and subtract that as our next step

             x^2  -5x
     -----------------------------
x+5  | x^3  + 0x^2 + 0x  + 125
           x^3  + 5x^2
         --------------------
          0x^3 - 5x^2 + 0x + 125
                    -5x^2 -25x
                    ----------------
                      0x^2 + 25x + 125

Now, what can we multiply with $x+5$ to get $25x + 125$?

whpalmer4 · Answer

no, this is long division

whpalmer4 · Answer

so what do you have to multiply with (x+5) to get 25x + 125?

anonymous · Answer

5x+25

anonymous · Answer

right?

whpalmer4 · Answer

no, not quite.   (5x+25)(x+5) = 5x^2 + 25x + 25x + 125

how about just 25?

25(x+5) = 25x + 5*25 = 25x + 125

whpalmer4 · Answer

so that means $$\frac{x^3+125}{x+5} = x^2-5x+25$$

and we can factor $$x^3+125 = (x+5)(x^2-5x+25)$$

Can we factor $x^2-5x+25$ some more?

anonymous · Answer

no

anonymous · Answer

i put that into the calculator and i just got 25, @whpalmer4

whpalmer4 · Answer

you put what into the calculator?

anonymous · Answer

no i get it now, its okay. thanks

whpalmer4 · Answer

we can't factor$ x^2-5x+25$ any more because the only factors we have to work with are either 1 and 25, or 5 and 5, and there's no combination of them that we could get to multiply to 25 and sum to -5.  Therefore, it is fully factored.