How to Factorize x^2-y^2-x+y: A Guide for Solving Polynomial Equations

[mathlearn]

Factorise \(\displaystyle x^2-y^2-x+y\). Any Ideas on how to begin (Mmm)

 

[MarkFL]

If you observe that:

\(\displaystyle x^2-y^2=(x+y)(x-y)\)

and

\(\displaystyle -x+y=-(x-y)\)

then can you proceed to factor?

 

[mathlearn]

If you observe that:

\(\displaystyle x^2-y^2=(x+y)(x-y)\)

and

\(\displaystyle -x+y=-(x-y)\)

then can you proceed to factor?

Thanks (Yes)

so now to factor (x+y)(x-y)-(x-y)

Can this be factorised any further ? Agree ?(Thinking)

Many Thanks (Happy)

 

[mrtwhs]

Thanks (Yes)

so now to factor (x+y)(x-y)-(x-y)

Can this be factorised any further ? Agree ?(Thinking)

Many Thanks (Happy)

When you have an expression of 4 or more terms, a good strategy is the group some of them together. This problem has 4 terms. One try would have been to group 3 together and leave one by itself. Another approach (as hinted by MarkFL) is to group 2 together and 2 together. You have done this and arrived at \(\displaystyle (x+y)(x-y)-(x-y)\). Now ask yourself how you would factor \(\displaystyle ab-b\).

 

[mathlearn]

When you have an expression of 4 or more terms, a good strategy is the group some of them together. This problem has 4 terms. One try would have been to group 3 together and leave one by itself. Another approach (as hinted by MarkFL) is to group 2 together and 2 together. You have done this and arrived at \(\displaystyle (x+y)(x-y)-(x-y)\). Now ask yourself how you would factor \(\displaystyle ab-b\).

Hope this is the answer $ x^2-y^2-x+y=(x^2-y^2)-(x-y)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

But I am not exactly clear on how did $(x-y)(x+y)-(x-y)$ become $(x-y)(x+y-1)$, Apologies because I am not that much good at factoring. (Crying)

Now ask yourself how you would factor \(\displaystyle ab-b\).

I know that this is the case, but can someone explain it a little , replacing the ab-b with the relevant terms.

If possible can a resource on factorization be posted.

Many Thanks

 

[HallsofIvy]

Hope this is the answer $ x^2-y^2-x+y=(x^2-y^2)-(x-y)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

But I am not exactly clear on how did $(x-y)(x+y)-(x-y)$ become $(x-y)(x+y-1)$, Apologies because I am not that much good at factoring. (Crying)
I know that this is the case, but can someone explain it a little , replacing the ab-b with the relevant terms.

If possible can a resource on factorization be posted.

Many Thanks

Do you know what "factorization" means? Did you recognize that ab- b= (a- b)b?
Compare ab- b to (x+ y)(x- y)- (x- y). What do you think "a" and "b" are in terms of x and y?

 

[kaliprasad]

Do you know what "factorization" means? Did you recognize that ab- b= (a- b)b?
Compare ab- b to (x+ y)(x- y)- (x- y). What do you think "a" and "b" are in terms of x and y?

there is a typo error above ab-b =(a-1)b

 

[HallsofIvy]

there is a typo error above ab-b =(a-1)b

Oops! Thanks!

 

[mathlearn]

$x^2-y^2-x+y=(x^2-y^2)-(x-y)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

$(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

As there are two $(x-y)$ , take it once and now a '-1' is isolated,

$(x-y)(x+y)-(x-y) = (x-y)(x+y)-1 = (x-y)(x+y-1)$

Correct?

 

[MarkFL]

$x^2-y^2-x+y=(x^2-y^2)-(x-y)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

This is correct.

$(x-y)(x+y)-(x-y) = (x-y)(x+y)-1 = (x-y)(x+y-1)$

This is not correct. This should be:

$(x-y)(x+y)-(x-y) = (x-y)((x+y)-1) = (x-y)(x+y-1)$

 

[mathlearn]

This is correct.
This is not correct. This should be:

$(x-y)(x+y)-(x-y) = (x-y)((x+y)-1) = (x-y)(x+y-1)$

Can you explain a little bit on what happens there in words. (Smile)

 

[MarkFL]

Look at the difference between what you posted and what I posted. You simply did not use correct bracketing symbols...you left that $-1$ dangling out there by itself. You essentially stated that:

\(\displaystyle ab-a=ab-1\)

when what we want is:

\(\displaystyle ab-a=a(b-1)\)