(xy+1)(x+1)(y+1)+xy 因式分解

Jasonbradley

4 min read Jun 17, 2024

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

This article will guide you through the process of factoring the expression (xy+1)(x+1)(y+1)+xy.

Expanding the Expression

First, we need to expand the expression by multiplying out the brackets.

Let's start by multiplying the first two brackets:

(xy+1)(x+1) = xy(x+1) + 1(x+1) = x²y + xy + x + 1

Now, let's multiply this result by (y+1):

(x²y + xy + x + 1)(y+1) = x²y(y+1) + xy(y+1) + x(y+1) + 1(y+1) = x²y² + x²y + xy² + xy + xy + x + y + 1

Simplifying this, we get:

x²y² + x²y + xy² + 2xy + x + y + 1

Finally, let's add the term xy back in:

x²y² + x²y + xy² + 2xy + x + y + 1 + xy

This gives us the expanded form of our original expression.

Factoring by Grouping

We can now factor this expression by grouping terms. Notice that we can group the first four terms together and the last three terms together:

(x²y² + x²y + xy² + 2xy) + (x + y + 1)

Now, let's factor out common factors from each group:

xy(xy + x + y + 2) + (x + y + 1)

We can see that the expression in the first group is very close to the expression in the second group. To make them identical, we can factor out a 1 from the second group:

xy(xy + x + y + 2) + 1(x + y + 1)

Now we have two identical expressions in the brackets:

(xy + 1)(xy + x + y + 2)

Final Factoring

The last step is to factor the second expression in the brackets by grouping. We can group the first two terms and the last two terms:

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

Finally, we can factor out the common factor (y+1):

(xy + 1)(y+1)(x+1)

Conclusion

We have successfully factored the expression (xy+1)(x+1)(y+1)+xy into (xy + 1)(y+1)(x+1). This process involves expanding the expression, grouping terms, factoring out common factors, and simplifying.