Factor by Grouping: spent 15-20 mins on diff. arrangements for 28u + ux - 4x - 7u^2

[dnymeyer]

I have a question about factoring by grouping.

First, I do understand factoring is about finding GCF and using distributive property in reverse.
My question is not about the mechanics of factoring by grouping, but rather how to be more efficient in situations where rearranging the terms is necessary.

For example:

28u + ux - 4x - 7u^2

If you try to factor these terms without rearranging, there is really no room to group the terms to completion. So I understand it is necessary to rearrange the terms.

I understand that for this particular problem, you need to rearrange the terms in this way in order to get an answer:

28u - 4x + ux -7u^2

Answer: (7u - x)(4-u)


This answer took me 15-20 minutes to figure out, because I spent so much time guessing on the arrangement of the terms.

My question... is there a more efficient way to find the arrangement of the terms? As far as I can see, it is truly just guessing and trying to solve and guessing again and again until you find a way which you are able to solve. Any tips?

[EDIT] I just realized it may not be clear why I am rearranging the terms. I am solving the equation by first grouping into two terms, and then solving for GCF of both terms. As per the example, after rearranging, I solved like this:

28u - 4x +ux - 7u^2
(28u - 4x) + (ux - 7u^2)
4(7u-x) - u(-x + 7u)
4(7u - x) - u(7u - x)

Answer: (7u - x)(4 - u)

 

[Steven G]

I have a question about factoring by grouping.

First, I do understand factoring is about finding GCF and using distributive property in reverse.
My question is not about the mechanics of factoring by grouping, but rather how to be more efficient in situations where rearranging the terms is necessary.

For example:

28u + ux - 4x - 7u^2

If you try to factor these terms without rearranging, there is really no room to group the terms to completion. So I understand it is necessary to rearrange the terms.

I understand that for this particular problem, you need to rearrange the terms in this way in order to get an answer:

28u - 4x + ux -7u^2

Answer: (7u - x)(4-u)


This answer took me 15-20 minutes to figure out, because I spent so much time guessing on the arrangement of the terms.

My question... is there a more efficient way to find the arrangement of the terms? As far as I can see, it is truly just guessing and trying to solve and guessing again and again until you find a way which you are able to solve. Any tips?

[EDIT] I just realized it may not be clear why I am rearranging the terms. I am solving the equation by first grouping into two terms, and then solving for GCF of both terms. As per the example, after rearranging, I solved like this:

28u - 4x +ux - 7u^2
(28u - 4x) + (ux - 7u^2)
4(7u-x) - u(-x + 7u)
4(7u - x) - u(7u - x)

Answer: (7u - x)(4 - u)

Hi, I am not that clear why it takes you so long to guess. There are just four terms, so try first grouping the 1st and 2nd terms together (so as a consequence of that grouping you also grouped the 3rd and 4th term together). If that doesn't work then try the 1st and the 3rd terms, if that fails then try grouping the 1st and the 4th terms. There are only three possible ways to group.
There is a quicker way (some will say it the same way I just mentioned). Look at the 1st two terms and just observe what is common and see what is left over after you factor the gcd. Keep the left over in your head or right it down. Then see what is left over after you factor the gcd from the 3rd and 4th terms. If what is left over matches then you found the correct grouping. If not then try grouping the 1st and 3rd terms AND the 2nd and 4th terms and see what is left over. If this did not work then group the 1st and 4th terms AND group the 2nd and 3rd terms. If this too fails then you can not factor by grouping.

In 28u + ux - 4x - 7u^2 I see the coefficients 28, 1, 4 and 7. I immediately see, because I am good at this from doing many, 7 is a factor of 28 and 7 and after factoring out a 7 I will be left with 1 and 4 WHICH are the other two numbers. To be quick at this you need to see what I saw. Do many of these problems and try to see which terms need to be grouped together. It will be easy after a while. Good luck and great question.

 

[Dr.Peterson]

I have a question about factoring by grouping.

First, I do understand factoring is about finding GCF and using distributive property in reverse.
My question is not about the mechanics of factoring by grouping, but rather how to be more efficient in situations where rearranging the terms is necessary.

For example:

28u + ux - 4x - 7u^2

If you try to factor these terms without rearranging, there is really no room to group the terms to completion. So I understand it is necessary to rearrange the terms.

I understand that for this particular problem, you need to rearrange the terms in this way in order to get an answer:

28u - 4x + ux -7u^2

Answer: (7u - x)(4-u)


This answer took me 15-20 minutes to figure out, because I spent so much time guessing on the arrangement of the terms.

My question... is there a more efficient way to find the arrangement of the terms? As far as I can see, it is truly just guessing and trying to solve and guessing again and again until you find a way which you are able to solve. Any tips?

[EDIT] I just realized it may not be clear why I am rearranging the terms. I am solving the equation by first grouping into two terms, and then solving for GCF of both terms. As per the example, after rearranging, I solved like this:

28u - 4x +ux - 7u^2
(28u - 4x) + (ux - 7u^2)
4(7u-x) - u(-x + 7u)
4(7u - x) - u(7u - x)

Answer: (7u - x)(4 - u)

There is a very good trick for quickly finding a workable order of terms, if they don't factor as given.

Consider this example, where I am starting with the factored form and expanding to get a polynomial you might be assigned to factor:

(2x + 3)(3x - 5) = 6x^2 - 10x + 9x - 15 = 6x^2 - x - 15

I started with two factors both written in descending order, and ended up with a polynomial in descending order. This will always happen.

So if the terms you are given are not in descending order, make them so, and, if factoring by grouping can work at all, it will! (This also ensures that the factors will conveniently be in descending order.) If we were given 9x + 6x^2 - 10x - 15, we would rearrange as 6x^2 - 10x + 9x - 15.

When there are more than one variable, you can take this either in terms of the total degree of each term (that is, the sum of the exponents on the variables), or in terms of just one of the variables (as if the others were just numbers).

Now take your example:

28u + ux - 4x - 7u^2

The degrees of the terms are, respectively, 1, 2, 1, 2. To follow my rule, we want to put the 2's before the 1's. That leaves some ambiguity; which of ux and -7u^2 should come first? To deal with that, I then focus on one variable, say the x. In terms of x, the degrees of the terms are 0, 1, 1, 0. This distinguishes the terms sufficiently.

So we start with ux and -7u^2, in that order because the degrees in x are 1, 0 respectively. Then we follow those with 28u and -4x, but reverse their order so the term with x comes first. This gives us:

ux - 7u^2 - 4x + 28u

Now we factor:

u(x - 7u) - 4(x - 7u)
(u - 4)(x - 7u)

I could just as well have thought of u as the variable to focus on; then I would have

-7u^2 + ux + 28u - 4x

I find it safer to have a positive leading term (as negatives can lead us astray here), so I would first factor out -1:

-1[7u^2 - ux - 28u + 4x]
-1[u(7u - x) - 4(7u - x)]
-1(u - 4)(7u - x)

This is equivalent to the other, but would be easier to work with if, subsequently, we needed to solve for u.

(By the way, nothing you did is solving an equation -- there are no equations here!)

 

[dnymeyer]

Dr. Peterson,

Very clear information and a great trick. I was not taught this method. I was taught the "guess and check" method, which is tedious and frustrating.

I will be using this trick from now on! Thank you!