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Chapter 5: Problem 74

Factor each expression. $$ 8 x^{6}-27 y^{3} $$

Short Answer

Expert verified
The expression factors to \((2x^2 - 3y)(4x^4 + 6x^2y + 9y^2)\).

Step by step solution

01

Identify the Structure

The expression \(8x^6 - 27y^3\) resembles a difference of cubes. Recognize that \(8x^6 = (2x^2)^3\) and \(27y^3 = (3y)^3\). This matches the formula for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = 2x^2\) and \(b = 3y\).
02

Apply the Difference of Cubes Formula

Using \(a = 2x^2\) and \(b = 3y\), substitute into the difference of cubes formula.\[(2x^2 - 3y)((2x^2)^2 + (2x^2)(3y) + (3y)^2)\]
03

Simplify the Factored Expression

Calculate \((2x^2)^2\), \((2x^2)(3y)\), and \((3y)^2\).\[(2x^2 - 3y)(4x^4 + 6x^2y + 9y^2)\]

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Key Concepts

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

Difference of Cubes
When you're learning about factoring polynomials, one of the special identities you encounter is the "difference of cubes". This refers to expressions that take the form of \(a^3 - b^3\).
The difference of cubes is one of the identities that allow us to factor polynomials more easily and is highly useful in simplifying complex algebraic expressions.
To factor a difference of cubes, we use the formula:
  • Formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
In this formula:
  • \(a^3\) is the cube of a number \(a\)
  • \(b^3\) is the cube of another number \(b\)
By identifying \(a\) and \(b\) from the given expression, you can apply this formula to factor the expression into a product of two simpler polynomials. Understanding this identity is crucial because it simplifies expressions and helps solve algebraic equations.
Polynomial Expressions
Polynomial expressions are algebraic expressions that consist of variables and coefficients involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental in algebra because they appear in a range of problems from the simplest equations to the most complex algebraic structures.
Let's understand the basic structure:
  • Terms: A polynomial is made up of terms, which can be a constant, a variable, or a combination of both.
  • Degree: The largest exponent present in the polynomial determines its degree, indicating the highest power of the variable.
  • Coefficient: The numerical factor of a term. In \(8x^6\), 8 is the coefficient.
When dealing with polynomials, recognizing their degree and the form they take (such as cubes or squares) plays a significant role in how they can be manipulated, especially when factoring or simplifying complex expressions.
Algebraic Identities
Algebraic identities are true for all values of the variables and are extremely useful tools in simplifying and factoring polynomial expressions. These identities, such as the difference of squares or cubes, provide shortcuts to break down complex problems into manageable steps.
For instance:
  • Common Identity: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
  • Usage: They help in quickly recognizing patterns within polynomial expressions, reducing them to a more straightforward form.
By applying algebraic identities, you can transform an expression like \(8x^6 - 27y^3\) into a factored form using the steps laid out in the difference of cubes identity.
Understanding these identities enhances problem-solving skills and helps in areas of mathematics that require simplification or factorization.

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Most popular questions from this chapter

Graph the solution set of each inequality or system of inequalities on a rectangular coordinate system. $$ \left\\{\begin{array}{l} y-2<3 x \\ y+2 x<3 \end{array}\right. $$

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Factor Assume that \(n\) is a natural number. $$ x^{2 n}+2 x^{n}+1 $$

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Explain why the factorization is not complete. $$ \text { Factor: } \quad 1-t^{8}=\left(1+t^{4}\right)\left(1-t^{4}\right) $$
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