Problem 61 Factor $$ x^{6 a}-\left(x^{2... [FREE SOLUTION] (2024)

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Chapter 5: Problem 61

Factor $$ x^{6 a}-\left(x^{2 a}+1\right)^{3} $$

Short Answer

Expert verified

The factored form is \[ - (3x^{4a} + 3x^{2a} + 1)\]

Step by step solution

- Identify the expression

The given expression is \[ x^{6a} - (x^{2a} + 1)^{3} \]. Notice that this can be recognized as a difference of cubes.

- Recall the difference of cubes formula

The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. In our case, identify $a$ and $b$ such that $a = x^{2a}$ and $b = x^{2a} + 1$.

- Apply the formula

Set $a = x^{2a}$ and $b = x^{2a} + 1$. Using the difference of cubes formula, we get: \[ (x^{2a})^3 - (x^{2a} + 1)^3 = \] \[ (x^{2a} - (x^{2a} + 1))((x^{2a})^2 + x^{2a}(x^{2a} + 1) + (x^{2a} + 1)^2) \]

- Simplify the expressions

Simplify the terms step-by-step: \[ x^{2a} - (x^{2a} + 1) = x^{2a} - x^{2a} - 1 = -1 \] Now, for the remaining terms: \[ (x^{2a})^2 + x^{2a}(x^{2a} + 1) + (x^{2a} + 1)^2 = x^{4a} + x^{4a} + x^{2a} + x^{4a} + 2x^{2a} + 1 \]

- Combine simplified terms

Combine the simplified terms:\[ -1((x^{4a} + x^{2a} + 1) + x^{4a} + 2x^{2a} + x^{4a}) = -1 (3x^{4a} + 3x^{2a} + 1) \]

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring

Factoring is the process of breaking down an expression into simpler terms or factors that, when multiplied together, produce the original expression. In algebra, factoring is crucial because it simplifies polynomial expressions, making them easier to solve or manipulate. For instance, if we have the polynomial , we look for common terms or use algebraic identities to simplify it. In the given exercise, we recognize the expression

algebraic identities

Understanding algebraic identities can simplify working with polynomial expressions. One crucial algebraic identity used in this exercise is the difference of cubes formula. The difference of cubes states that:

polynomial expressions

Polynomial expressions are algebraic expressions consisting of terms in the form of Formular-based approach to recognizing and simplifying polynomial expressions helps solve complex problems. For instance, Note, understanding and using polynomial expression concepts not only helps in academics but also in real-world applicable areas like engineering and economics. Use the knowledge gained from factoring and applying algebraic identities to master the simplification and manipulation of polynomial expressions!

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$Problem 61 Factor $$ x^{6 a}-\left(x^{2... [FREE SOLUTION] (3)$

Most popular questions from this chapter

Factor completely. $$ a^{4}-50 a^{2} b^{2}+49 b^{4} $$Perform the indicated operations. $$ \left(x^{2}-4 x+7\right)+\left(3 x^{2}-9\right)-\left(x^{2}-4 x+7\right) $$Perform the indicated operation. $$ \left(2 x^{2 a}+4 x^{a}+3\right)+\left(6 x^{2 a}+3 x^{a}+4\right) $$

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