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Chapter 15: Problem 388

Factor \(128 \mathrm{x}^{6}-2 \mathrm{y}^{6}\)

Short Answer

Expert verified
The short answer is: \(2(8x^3 + y^3)(8x^3 - y^3)\)

Step by step solution

01

Find the Greatest Common Factor (GCF)

The given expression is \(128x^6 - 2y^6\). In order to find the GCF, we need to check factors of both terms. - The factors of \(128x^6\) are: 2, 64, 128, x, \(x^2\), \(x^3\), \(x^4\), \(x^5\), and \(x^6\). - The factors of \(2y^6\) are: 2, y, \(y^2\), \(y^3\), \(y^4\), \(y^5\), and \(y^6\). The GCF of both terms is 2.
02

Factor the GCF

Since the GCF is 2, we need to factor this out. Divide both terms by 2: \[\frac{128x^6}{2} = 64x^6\] \[\frac{-2y^6}{2} = -y^6\] Now, rewrite the given expression with the factored GCF: \(2(64x^6 - y^6)\)
03

Recognize the difference between two squares

Now, we need to look for a pattern inside the parentheses. We can see that the expression inside the parentheses is a difference between two squares: \(64x^6 - y^6 = (8x^3)^2 - (y^3)^2\)
04

Factor the difference between two squares

The difference between two squares can be factored as follows: \(a^2 - b^2 = (a+b)(a-b)\). Applying this to our expression: \((8x^3)^2 - (y^3)^2 = (8x^3 + y^3)(8x^3 - y^3)\) Now, combine this with the GCF we factored out earlier: \(2(8x^3 + y^3)(8x^3 - y^3)\)
05

Simplify the factored expression (if needed)

In this case, there is no need to further simplify the factored expression. Our final factored expression is: \(2(8x^3 + y^3)(8x^3 - y^3)\)

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Most popular questions from this chapter

Reduce to lowest form: \(a^{3}-8 b^{3} / 2 a^{2}-8 b^{2}\).

Combine into a single fraction \(\left(2 \mathrm{x} / \mathrm{x}^{2}-6 \mathrm{x}+9\right)-\left(8 / \mathrm{x}^{2}-2 \mathrm{x}-3\right)-(1 / \mathrm{x}+1)\)

Factor \(27 \mathrm{x}^{3}-8 \mathrm{y}^{3}\)

Factor \(\mathrm{x}^{3}-8\)

Factor x2 + 7x + 10 .
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