(xy-1)(x-1)(y+1)-xy Factor

2 min read Jun 17, 2024
(xy-1)(x-1)(y+1)-xy Factor

Factoring (xy-1)(x-1)(y+1)-xy

This expression can be factored to reveal a simpler form. Let's break down the steps:

Step 1: Expand the expression

Begin by expanding the first three terms of the expression:

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

Further simplification:

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

Step 2: Combine like terms

Now combine all the similar terms:

x^2y^2 + x^2y - xy^2 - 3xy - x + y + 1

Step 3: Factor by grouping

Group the terms to facilitate factoring:

(x^2y^2 - xy^2) + (x^2y - 3xy) + (-x + y + 1)

Factor out common terms from each group:

xy^2(x-1) + xy(x-3) + (-x + y + 1)

Step 4: Recognize a common factor

Notice that the last three terms are the negative of the first three terms. We can rewrite them by factoring out a -1:

xy^2(x-1) + xy(x-3) - (x - y - 1)

Now we have a common factor of (x-1) in the first two terms and a common factor of -(x-1) in the last term:

(x-1)(xy^2 + xy -1)

Final Factored Form

Therefore, the factored form of the expression is:

(x-1)(xy^2 + xy -1)

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