MHB How to Factorize x^2-y^2-x+y for Solving Polynomial Equations

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The discussion focuses on factorizing the expression x^2 - y^2 - x + y. Participants identify that x^2 - y^2 can be factored as (x + y)(x - y) and that -x + y can be rewritten as -(x - y). The expression is simplified to (x + y)(x - y) - (x - y), leading to the factorization (x - y)(x + y - 1). Clarifications are sought on the steps taken to reach this final form, particularly regarding the manipulation of terms and the importance of correct bracketing in factorization. The conversation emphasizes the need for clear explanations and resources on factoring techniques.
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Factorise $$x^2-y^2-x+y$$. Any Ideas on how to begin (Mmm)
 
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If you observe that:

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

and

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

then can you proceed to factor?
 
MarkFL said:
If you observe that:

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

and

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

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)
 
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mathlearn said:
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 $$(x+y)(x-y)-(x-y)$$. Now ask yourself how you would factor $$ab-b$$.
 
mrtwhs said:
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 $$(x+y)(x-y)-(x-y)$$. Now ask yourself how you would factor $$ab-b$$.

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)

mrtwhs said:
Now ask yourself how you would factor $$ab-b$$.

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 :rolleyes:
 
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mathlearn said:
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 :rolleyes:
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?
 
HallsofIvy said:
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
 
kaliprasad said:
there is a typo error above ab-b =(a-1)b
Oops! Thanks!
 
$x^2-y^2-x+y=(x^2-y^2)-(x-y)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

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

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?
 
  • #10
mathlearn said:
$x^2-y^2-x+y=(x^2-y^2)-(x-y)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1)$

This is correct.

mathlearn said:
$(x-y)(x+y)-(x-y) = (x-y)(x+y)-1 = (x-y)(x+y-1)$

This is not correct. This should be:

$(x-y)(x+y)-(x-y) = (x-y)((x+y)-1) = (x-y)(x+y-1)$
 
  • #11
MarkFL said:
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)
 
  • #12
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:

$$ab-a=ab-1$$

when what we want is:

$$ab-a=a(b-1)$$
 
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