Factor by grouping: 18r^2 - 2ty + 12ry - 3rt

[sean1]

I can only get so far before my answer comes out all mixed up. I am afraid I missed a step when the professor was teaching today. This is what I have so far:

Factor by grouping.
18r^2 - 2ty + 12ry - 3rt

In class we were told to rewrite it like this since GSM is 1
(18r^2 - 2ty) + (12ry - 3rt)

From this I've worked out that the GCF is 6r+t
?(6r+t) +?(6r+t)
I am not sure where to go from here. Anything I do would just be guessing and I am hoping someone can break down the correct thing I should do at this point.

Thank you for making it this far ?

 

[Steven G]

I can only get so far before my answer comes out all mixed up. I am afraid I missed a step when the professor was teaching today. This is what I have so far:

Factor by grouping.
18r^2 - 2ty + 12ry - 3rt

In class we were told to rewrite it like this since GSM is 1
(18r^2 - 2ty) + (12ry - 3rt)

From this I've worked out that the GCF is 6r+t
?(6r+t) +?(6r+t)
I am not sure where to go from here. Anything I do would just be guessing and I am hoping someone can break down the correct thing I should do at this point.

Thank you for making it this far ?

Can you rethink about the GCF (18r^2 - 2ty , 12ry - 3rt) = 6r +t.

You should try to factor each set of brackets separately

Factor 18r^2 - 2ty :
18r^2 - 2ty = 3*3*2*r*r - 2*t*y. Clearly just a 2 is in common (ie the GCF (18r^2 , 2ty) =2). So 18r^2 - 2ty = 2(9r^2-ty).

Now factor (12ry - 3rt)
(12ry - 3rt) = (4*3*r*y - 3*r*t) = 3r(4y - t).

Now since (9r^2-ty) and (4y - t) are different this factoring was NOT helpful.

Now you have 4 terms, namely 18r^2 , 2ty, 12ry and 3rt. Try a different ordering and then try again. I would try (18r^2 -3rt) + (12ry - 2ty)

 

[stapel]

Factor by grouping.
18r^2 - 2ty + 12ry - 3rt

In class we were told to rewrite it like this since GSM is 1
(18r^2 - 2ty) + (12ry - 3rt)

From this I've worked out that the GCF is 6r+t
?(6r+t) +?(6r+t)
I am not sure where to go from here.

I have no idea where to go from there, either. Instead, try working "in pairs".

In the pairs of terms, as written, there's nothing much you can do with the first pair:

. . . . .\(\displaystyle 18r^2\, -\, 2ty\, =\, 2\, (9r^2\, -\, ty)\)

And the remaining pair factors as:

. . . . .\(\displaystyle 12ry\, -\, 3rt\, =\, 3r\, (4y\, -\, t)

...which doesn't match with the first pair.

So this is probably not the way that you're expected to factor. Instead, look for a term which might share more with the first term; I'll try the third term, and see if that grouping works a little better:

. . . . .\(\displaystyle (18r^2\, +\, 12ry)\, +\, (-2ty\, -\,3rt)\)

. . . . .\(\displaystyle 18r^2\, +\, 12ry\, =\, 6r\, (3r\, +\, 2y)\)

. . . . .\(\displaystyle -2ty\, -\, 3rt\, =\, -t\, (2y\, +\, 3r)\)

Well, the order of the terms in the parentheticals is different, but flip one of them around, and you're golden. Also, this might not be the only way to obtain a helpful grouping. What if I try the fourth term with the first?

. . . . .\(\displaystyle (18r^2\, -\, 3rt)\, +\, (-2ty\, +\, 12ry)\)

. . . . .\(\displaystyle 18r^2\, -\, 3rt\, =\, 3r\, (6r\, -\, t)\)

. . . . .\(\displaystyle -2ty\, +\, 12ry\, =\, 2y\, (-t\, +\, 6r)\)

So this grouping (after a bit of rearranging) will also work. Moral: If one grouping doesn't work, try another! \)

 

[Dr.Peterson]

I can only get so far before my answer comes out all mixed up. I am afraid I missed a step when the professor was teaching today. This is what I have so far:

Factor by grouping.
18r^2 - 2ty + 12ry - 3rt

In class we were told to rewrite it like this since GSM is 1
(18r^2 - 2ty) + (12ry - 3rt)

From this I've worked out that the GCF is 6r+t
?(6r+t) +?(6r+t)
I am not sure where to go from here. Anything I do would just be guessing and I am hoping someone can break down the correct thing I should do at this point.

To supplement what others have said, I find it helpful, when factoring by grouping, to arrange the terms in descending order, by either the total degree of each term (sum of exponents of variables) or with respect to one variable. Here, all terms have total degree 2; but focusing on the r, the terms 18r^2 - 2ty + 12ry - 3rt have degrees 2, 0, 1, 1, suggesting that you move -2ty to the last position. This turns out to work well.

I am curious, though: what does GSM mean? Was that just a typo for "GCF" (of all terms)?

 

[Denis]

Well, to factor: 18r^2 - 2ty + 12ry - 3rt
seems evident that we'll end up with something like:
(?r +- ?)(?r +- ?)

Note: 9 words used