Find the greatest common factor of -30x^4 yz^3 and 75x^4 z^2.

Question

whpalmer4 · Answer

$$GCF(-30x^4yz^3,75x^4z^2)$$

Okay, you can do this piece by piece and multiply the components.  What is the GCF of 30 and 75?

anonymous · Answer

15

whpalmer4 · Answer

Good!  What's the GCF of $x^4$ and $x^4$? :-)

anonymous · Answer

x^4

whpalmer4 · Answer

Correct again.  Now the $y$ portion only appears in 1 of the items, so it doesn't contribute to the GCF.  What is the GCF of $z^3$ and $z^2$?

anonymous · Answer

z^2

whpalmer4 · Answer

Correctamundo.  So now we multiply all of the components we found together.  What's the answer?

anonymous · Answer

15x^4z^2

whpalmer4 · Answer

That's correct. 

Another way you could think of this is factoring.  Say we had $$-30x^4 yz^3 + 75x^4 z^2$$and wanted to write it in the form $a(b+c)$.  What could we factor out to use as $a$? That's the GCF!

We could rewrite it as $$15x^4z^2(-2yz+5)$$right?

anonymous · Answer

Yes. I use factoring a lot in this unit of algebra:)

whpalmer4 · Answer

If you want a more formal description, to find the GCF, we factor each of the items into their prime components:

$$-30x^4yz^3= -1*2*3*5*x^4*y*z^3$$$$75x^4z^2 = 3*5^2*x^4*z^2$$
Now we collect the highest common powers of each of the common factors and multiply them together:
$$GCF = 3*5*x^4*z^2 = 15x^4z^2$$