How to Factorize Complex Expressions Involving Polynomials

When you stumble upon a complex polynomial expression such as x^2 2x - 8 - y^2 6y, it can appear intimidating at first glance. However, by breaking down the expression into manageable parts and using algebraic techniques like completing the square and factoring differences of squares, you can factorize it effectively. In this article, we will walk through the process step by step and provide a clear, detailed explanation of how to handle such expressions.

Understanding the Expression

The given expression is x^2 2x - 8 - y^2 6y. To simplify this, we will first rearrange and group the terms for x and y separately.

Rearranging the terms:

x^2 2x - 8 - y^2 6y

Factoring the x Terms

Let's focus on the first part of the expression: x^2 2x - 8.

Step 1: Identify two numbers that multiply to -8 and add to 2. The numbers are 4 and -2.

Step 2: Rewrite the expression using these numbers:

x^2 2x - 8 (x 4)(x - 2)

Factoring the y Terms

Now, let's look at the second part of the expression: - y^2 6y.

Step 1: Complete the square for - y^2 6y.

- y^2 6y - (y^2 6y) - (y - 3)^2 9 - (y - 3)(y - 3) 9

Step 2: Simplify and combine the terms:

- (y - 3)^2 9 - (y - 3)^2 9

Combining Everything

NOW, substituting back into our expression:

x^2 2x - 8 - y^2 6y (x 4)(x - 2) - (y - 3)^2 9

Simplifying further:

x^2 2x - 8 - y^2 6y (x 4)(x - 2) - (y - 3)^2 9

Final factored form:

x^2 2x - 8 - y^2 6y (x 4)(x - 2) - (y - 3)^2 9

This can be rearranged as:

x^2 2x - 8 - y^2 6y (x 4)(x - 2) - (y - 3)^2 9

A More Simplified Form

If you prefer a simpler representation, you can leave it as:

x^2 2x - 8 - y^2 6y (x 4)(x - 2) - (y - 3)^2 9

This shows the relationship clearly between the x and y components.

Additional Factoring Techniques

Another approach to factorize the expression is to rearrange the terms:

x^2 - y^2 - 2x - 8 - 6y (x - y)(x - y) - 2x - 8 - 6y (x - y)(x - y) - 2(x - 4) - 3(y - 2)

This is another way to factorize parts of the expression. However, it does not fully factorize the entire expression as initially intended.

Final Conclusion

By breaking down the expression and using techniques such as completing the square and recognizing differences of squares, you can effectively factorize complex polynomial expressions. The key steps include:

Rearranging and grouping terms for specific variables Completing the square for quadratic expressions Factoring differences of squares

This approach not only simplifies the expression but also provides a clear understanding of the underlying algebraic relationships.