Factoring a Sum of Cubes: How to Factor 8x^6+64?

Thread starter tmlfan_179027
Start date Sep 10, 2009

In summary, factoring is the process of breaking down a mathematical expression into smaller parts to simplify and solve equations more easily. To factor an expression with a common factor, identify the factor and divide each term by it. Variables with exponents can also be factored using the rules of exponents. For example, 8x^6+64 can be factored as 8(x^2+2x+4)(x^4-2x^3+4x^2-8x+16) and further using the rule of a^2+b^2=(a+b)(a-b).

Sep 10, 2009

tmlfan_179027

How would you factor 8x^6+64? My text states that the answer is 8(x^2+2)(x^4-2x^2+4). How would you get to that?

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Sep 10, 2009

CRGreathouse

Science Advisor

Homework Helper

Factoring out the 8 is easy.

Once you have that, it's clear that you have a sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

1. What is factoring and why is it important in mathematics?

Factoring is the process of breaking down a mathematical expression into smaller parts that can be multiplied together to get the original expression. It is important in mathematics because it allows us to simplify complex expressions and solve equations more easily.

2. How do I factor an expression with a common factor?

The first step is to identify the common factor among all the terms in the expression. Then, divide each term by the common factor and write the result in parenthesis. This common factor can then be factored out, leaving behind the remaining terms.

3. Can I factor an expression if it has variables with exponents?

Yes, you can factor an expression with variables and exponents. The key is to use the rules of exponents to rewrite the expression in a form where the common factor can be identified and factored out.

4. How would I factor 8x^6+64?

The first step is to identify the common factor, which in this case is 8. We can rewrite the expression as 8(x^6+8). Then, we can further factor the expression inside the parenthesis using the rule of a^3+b^3=(a+b)(a^2-ab+b^2). The final factored form is 8(x^2+2x+4)(x^4-2x^3+4x^2-8x+16).

5. Can this expression be factored further?

Yes, the expression 8x^6+64 can be factored further. We can use the rule of a^2+b^2=(a+b)(a-b) to factor the quadratic expression inside the parenthesis. The final factored form is 8(x^2+2x+4)(x^2-2x+4)(x^2+4).