How to Factorize Complex Polynomials: A Step-by-Step Guide

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In this guide, we will explore the process of factorizing a complex polynomial, 2x4-22,561,2y2-18y12-344,678. This guide provides a step-by-step method, employing various algebraic techniques to simplify and factorize the expression effectively. Understanding these techniques not only helps in solving advanced algebraic problems but also serves as a foundation for more complex mathematical concepts.

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Initial Simplification and Coefficient Combination

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Our journey begins by simplifying the given expression through the combination of constants. First, we combine the numeric constants: 22,561,2 - 344,678 47,120. This simplification is crucial as it allows us to better understand the polynomial's structure. The expression becomes:

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2x4-22,120y2-18y12

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Notice that each coefficient is even, enabling us to factor out a 2. Therefore, the expression becomes:

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2(x4-21,060y2-9y12)

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Applying Algebraic Techniques for Factorization

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To further simplify the expression inside the parentheses, we use the difference of squares formula, which is expressed as: a2-b2(a b)(a-b).

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First, let's tackle x4-21,060y2. Recognize that x4 can be written as (x2)2 and 21,060y2 as (213y2)2. Using the difference of squares formula, we factorize:

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x4-21,060y2 (x2 213y)(x2-213y)2

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Now, let's consider -9y12. Recognize that -9y12 can be written as (-3y6)2. Using the difference of squares formula, we factorize:

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-9y12 (-3y6)2

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Combining these factorizations, we have:

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2(x4-21,060y2-9y12) 2((x2 213y)(x2-213y) (3y6)2)

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To simplify further, let's focus on the terms inside the parentheses:

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(x2 213y)(x2-213y)-3y6

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Since we are dealing with a polynomial, we focus on factors that can be simplified effectively. We notice that the terms -213y and -3y6 can also be combined as -3y7.

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To achieve a more simplified factorization, we use the factoring form:

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2(x2-y6)(x2 y6)(x 3y)(x-3y)

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Conclusion

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Through a series of algebraic transformations and the application of the difference of squares formula, we have successfully factorized the complex polynomial into its constituent factors. This process not only simplifies the expression but also reveals underlying structures that can be applied to solving more complex algebraic problems.

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Understanding how to factorize polynomials is essential for mastering advanced algebra and is a valuable skill in fields such as engineering, physics, and advanced mathematics. By practicing these techniques, you can develop a deeper understanding of algebraic principles and improve your problem-solving abilities.