How to Factorize the Expression 0.4x^256y

The process of factorizing an algebraic expression is a fundamental concept in algebra. It involves expressing a given polynomial as a product of simpler polynomials. In this detailed guide, we will explore the steps to factorize the expression 0.4x^256y.

Understanding the Expression

The given expression is:

[P frac{2}{5}x^4 256y^4]

This is a bivariate polynomial, with variables x and y. The expression consists of two terms: one containing x^4 and another containing y^4, each multiplied by a constant coefficient.

Step 1: Multiply by 10 to Simplify the Coefficients

For simplicity, let's multiply the entire expression by 10:

[10P 4x^4 2560y^4]

Step 2: Recognize the Structure as a Sum of Squares

The expression (4x^4 2560y^4) can be viewed as a sum of squares. However, it is not immediately clear how to factor this sum. Let's proceed step by step.

Step 3: Rewrite the Expression

We can rewrite the expression as:

[4x^4 2560y^4 2^2 x^2^2 1016 y^2^2]

Notice that (2^2 4) and (1016 4 cdot 254).

Step 4: Complete the Square for Each Term

We will now complete the square for each term individually. Start with (2x^2^2):

[2x^2^2 1016y^2^2 (2x^2 4sqrt{10}y^2)^2 - (4sqrt{10}xy)^2]

Here, we used the identity (a^2 2ab b^2 (a b)^2). The terms were chosen to match the structure of the expression.

Step 5: Recognize the Difference of Squares

Now we have the expression in the form of the difference of two squares:

[(2x^2 4sqrt{10}y^2)^2 - (4sqrt{10}xy)^2]

Step 6: Factor Using the Difference of Squares Formula

The difference of squares formula is (a^2 - b^2 (a - b)(a b)). Applying this formula:

[(2x^2 4sqrt{10}y^2 - 4sqrt{10}xy)(2x^2 4sqrt{10}y^2 4sqrt{10}xy)]

Step 7: Conclusion

The factorized form of the expression is:

[P frac{1}{10} left[(2x^2 4sqrt{10}y^2 - 4sqrt{10}xy)(2x^2 4sqrt{10}y^2 4sqrt{10}xy)right]]

This concludes our detailed step-by-step guide to factorizing the given expression.

Key Concepts:

Factorization: The process of breaking down a polynomial into simpler factors. Polynomial Factorization: Expressing a polynomial as the product of two or more polynomials. Algebraic Expression: A combination of symbols representing numbers and operations.