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Chapter 6: Problem 266

Factor completely. $$ 64 x^{3}+125 y^{3} $$

Short Answer

Expert verified
The factored form of

Step by step solution

01

Identify the Formula

Recognize that the given expression is a sum of cubes. It fits the formula for factoring sums of cubes: Identify each term as a cube: - The first term, - The second term,
02

Apply the Sum of Cubes Formula

Using the sum of cubes formula, This can be written as:
03

Write Final Factored Form

Plugging the identified values into the formula, The final factored form of the expression

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Key Concepts

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

sum of cubes
The 'sum of cubes' is a special algebraic expression where two cube terms are added together. It follows a specific formula: \text{Sum of Cubes Formula:} \text{\text{Sum of Cubes Formula:}} \text{a}^3 + \text{b}^3 = (\text{a} + \text{b})(\text{a}^2 - \text{a}\text{b} + \text{b}^2) To factor an expression like\(64x^3 + 125y^3\), first identify each 'cube'. Here,\(64x^3\)is\((4x)^3\)and\(125y^3\)is\((5y)^3\). Plug these terms into the sum of cubes formula to get the final factorized form. It simplifies the problem.
factoring algebraic expressions
Factoring algebraic expressions involves breaking down a complex expression into simpler 'factors' that, when multiplied together, give you the original expression. For instance, in the expression\(64x^3 + 125y^3\), the goal is to rewrite it using multiplication. Start by recognizing the form of the equation. Then, apply the appropriate formula, such as the sum of cubes formula. This turns the problem into a more manageable form and helps you to easily find the solutions.
polynomial identities
Polynomial identities are like mathematical 'shortcuts' that help you quickly recognize and simplify polynomials. For instance, identities like the sum of cubes and difference of cubes help you easily factor complex expressions. The sum of cubes identity is particularly useful for problems like \(64x^3 + 125y^3\). Knowing these identities allows you to effectively tackle polynomial problems without unnecessary complications. It also helps in verifying your work as you can check if the simplified form follows the identity rules.
algebraic formulas
Algebraic formulas are key tools in algebra for simplifying and solving expressions. They are pre-derived general rules like the quadratic formula or sum of cubes formula. Take the sum of cubes formula: \text{\text{Sum of Cubes Formula:}} \text{a}^3 + \text{b}^3 = (\text{a} + \text{b})(\text{a}^2 - \text{a}\text{b} + \text{b}^2) You can directly apply this to the expression \text{\text{Sum of Cubes Formula:}} \(64x^3 + 125y^3\). These formulas save time and ensure accuracy.

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Most popular questions from this chapter

What is the GCF of \(y^{4}, y^{5},\) and \(y^{10} ?\) Write a general rule that tells you how to find the GCF of \(y^{a}, y^{b},\) and \(y^{c}\)

In the following exercises, factor by grouping. $$ 9 p^{2}+12 p-15 p-20 $$

In the following exercises, factor each expression using any method. $$ 12 x^{2}+36 y-24 z $$

In the following exercises, factor by grouping. $$ r^{2}-3 r-r+3 $$

In the following exercises, factor by grouping. $$ 4 x^{2}-36 x-3 x+27 $$
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