Question

Factor the polynomial completely. \(y^3-8\)

Short answer

The factorized form of \(y^3 - 8\) is \((y - 2)(y^2 + 2y + 4)\).

Step-by-step solution

Identify the form

Recognize that the given expression \((y^3 - 8)\) can be written in the form of \((a^3 - b^3)\) which is the difference of cubes.

Identify a and b

In this case, we identify that \(a\) is \(y\) and \(b\) is 2, because \(2^3 = 8\).

Apply the Formula

Substitute \(a\) and \(b\) into the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) to factorize the given expression.

Simplify

Plugging \(y\) for \(a\) and \(2\) for \(b\) into the formula, we get: \((y - 2)(y^2 + 2y + 4)\). This is the factorized form of the given expression.

Key concepts

Difference of Cubes

The concept of the Difference of Cubes is a fundamental tool in algebra for simplifying expressions. When you encounter a polynomial like \(y^3 - 8\), it's crucial to recognize it as a difference of cubes. This technique involves breaking down an expression of the form \(a^3 - b^3\) into a product of simpler binomials and trinomials.
In our specific example, \(8\) is actually \(2^3\), so we can rewrite the expression as \(y^3 - 2^3\). The general formula is:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

This formula helps transform a complex polynomial into a factorable expression involving \(a\) and \(b\). Here, \(a\) is \(y\) and \(b\) is \(2\). Recognizing and applying this identity not only allows us to simplify equations but also aids in solving algebraic problems quickly.

Algebraic Identities

Algebraic identities like the Difference of Cubes are critical in mathematics as they provide a universal method to simplify and manipulate expressions. Algebraic identities are essentially predetermined forms or equations that help simplify complex algebraic problems.
Let's briefly discuss how these identities are used:

• Identities allow us to transform expressions into different, often simpler forms without changing their value.
• The Difference of Cubes is one such identity. It involves expressing a polynomial as a product of binomials and trinomials involving rational numbers or variables.

Mastering algebraic identities is like learning shortcuts in algebraic computations. For instance, knowing that \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) provides a clear pathway to factoring expressions arising in real-world and theoretical math problems smoothly.

Step-by-Step Math Solutions

Step-by-step math solutions are invaluable for understanding and solving complex problems systematically. In the case of factoring \(y^3 - 8\), breaking the solution into clear steps makes the process straightforward.
First, identify the expression form, recognizing it as a difference of cubes. This sets the stage for determining the components of the identity to use. Then:

• Identify variables: In our example, identify \(a = y\) and \(b = 2\), since \(2^3 = 8\).
• Apply the identity: Substitute these into the known formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
• Simplify: Carry out the substitutions and simplify, reducing the expression to \((y - 2)(y^2 + 2y + 4)\).

Each step is a piece of the puzzle, and keeping them clear ensures you understand the transformation from start to finish. Step-by-step instructions demystify intricate algebra, allowing you to confidently tackle similar problems in the future.