Factoring Difference of Cubes x^3 8 YouTube from www.youtube.com
Introduction
In algebra, factoring is the process of breaking down a polynomial into its simplest form. When factoring x³ - 8, we are trying to find the factors that, when multiplied together, give us the original polynomial. In this article, we will discuss the steps to factor x³ - 8.
Step 1: Identify the Difference of Cubes
The first step in factoring x³ - 8 is to recognize that it is a difference of cubes. A difference of cubes is a polynomial of the form a³ - b³, which factors into (a - b)(a² + ab + b²). In the case of x³ - 8, a is x and b is 2.
Step 2: Apply the Difference of Cubes Formula
Using the formula for difference of cubes, we can factor x³ - 8 as (x - 2)(x² + 2x + 4). This is the simplest form of the polynomial.
Step 3: Check for Common Factors
It is important to note that there may be common factors that can be factored out of the polynomial before applying the difference of cubes formula. For example, if x³ - 8 was written as 8 - x³, we could factor out a -1 to get -(x³ - 8). Then, we could apply the difference of cubes formula to get -(x - 2)(x² + 2x + 4).
Step 4: Simplify the Factors
Once the polynomial has been factored, it is important to simplify the factors. In the case of x³ - 8, we cannot simplify the factors any further.
Step 5: Check Your Answer
To check your answer, you can use the distributive property to multiply the factors back together. For x³ - 8, we can use (x - 2)(x² + 2x + 4) and distribute to get x³ + 2x² - 4x - 8. This is the original polynomial, so we know our factoring is correct.
Conclusion
Factoring x³ - 8 involves recognizing it as a difference of cubes and applying the formula (a - b)(a² + ab + b²). It is important to check for common factors and simplify the factors before checking your answer. Factoring is a fundamental skill in algebra and is used in many applications, such as solving equations and finding roots of polynomials.
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