Factorizing the Expression 3x - y^2 - 2x - y: A Step-by-Step Guide

When faced with an algebraic expression like 3x - y^2 - 2x - y, factorization is a powerful technique to simplify and solve it. In this guide, we will walk you through the process of factorizing the expression step-by-step, ensuring clarity and understanding at each stage.

The given expression is:

3x - y^2 - 2x - y

This can be rewritten by combining like terms:

3x - 2x - y^2 - y x - y^2 - y

Step 1: Introduction to Factorization

Factorization is the process of breaking down an algebraic expression into simpler expressions that, when multiplied together, give the original expression. In this case, we will use substitution to simplify and factor the expression.

Step 2: Substitution

Let's define a new variable:

z x - y

Substituting z into the expression:

3z^2 - 2z

Step 3: Factoring the Expression

Now, we can factor out the common term z from the expression:

z(3z - 2)

Substituting back z x - y, we get:

(x - y)(3(x - y) - 2)

This can be further simplified to:

(x - y)(3x - 3y - 2)

Step 4: Verification and Expansion (Optional)

To ensure the factorization is correct, we can expand the factored form:

(x - y)(3x - 3y - 2)

Expanding the expression:

3x^2 - 3xy - 2x - 3xy 3y^2 2y

Simplifying by combining like terms:

3x^2 - 3xy - 3xy 3y^2 - 2x 2y 3x^2 - 6xy 3y^2 - 2x 2y

We can see that this is equivalent to the original expression:

3x - y^2 - 2x - y

Conclusion

The fully factored form of the expression 3x - y^2 - 2x - y is:

(x - y)(3x - 3y - 2)

If you find this process challenging, you can always verify it by expanding the factored form. This method of factorization can be applied to a wide range of algebraic expressions.

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