What Is The Completely Factored Form Of 8x2 50

What is the Completely Factored Form of 8x² + 50?

Factoring expressions is a fundamental concept in algebra. It involves breaking down a mathematical expression into simpler components that, when multiplied together, yield the original expression. This process is crucial for solving equations, simplifying expressions, and understanding the underlying structure of mathematical relationships. This article delves into the complete factorization of the expression 8x² + 50, exploring various factoring techniques and offering a comprehensive understanding of the process.

Understanding Factoring

Before diving into the specifics of factoring 8x² + 50, let's establish a foundational understanding of the concept of factoring. Factoring is essentially the reverse process of expanding an expression. When we expand an expression, we multiply terms together. Factoring, conversely, involves finding the terms that, when multiplied, produce the original expression.

For example, if we expand (x + 2)(x + 3), we get x² + 5x + 6. Factoring x² + 5x + 6 would involve reversing this process to arrive back at (x + 2)(x + 3).

Several factoring techniques exist, including:

• Greatest Common Factor (GCF): This involves identifying the largest factor common to all terms in the expression and factoring it out.
• Difference of Squares: This applies to expressions of the form a² - b², which factors to (a + b)(a - b).
• Trinomial Factoring: This involves factoring quadratic trinomials (expressions of the form ax² + bx + c) into two binomial expressions.
• Grouping: This technique is useful for expressions with four or more terms, where terms can be grouped to reveal common factors.

Factoring 8x² + 50: A Step-by-Step Approach

The expression 8x² + 50 doesn't immediately lend itself to trinomial factoring or the difference of squares. However, the greatest common factor (GCF) method can be effectively applied.

Step 1: Identify the GCF

Observe that both 8x² and 50 are even numbers. This suggests that 2 is a common factor. Let's examine further:

• 8x² = 2 * 4 * x²
• 50 = 2 * 25

The greatest common factor between 8 and 50 is 2. Therefore, we can factor out a 2 from the expression:

8x² + 50 = 2(4x² + 25)

Step 2: Examining the Remaining Expression

The expression within the parentheses, 4x² + 25, is a binomial. We need to check if it can be factored further. It resembles a sum of squares (a² + b²), which, in contrast to the difference of squares, cannot be factored using real numbers. The sum of squares can only be factored using complex numbers, which introduce imaginary units (i, where i² = -1).

Therefore, (4x² + 25) is not factorable over the real numbers. It is a prime polynomial.

Step 3: The Completely Factored Form

Since we've exhausted all factoring techniques over real numbers, the completely factored form of 8x² + 50 is 2(4x² + 25).

Factoring with Complex Numbers (Advanced)

While the factorization above is complete over real numbers, we can explore factoring the expression using complex numbers.

Recall that the sum of squares can be factored as:

a² + b² = (a + bi)(a - bi)

Applying this to our expression, 4x² + 25:

• a² = 4x² => a = 2x
• b² = 25 => b = 5

Therefore, 4x² + 25 can be factored as:

4x² + 25 = (2x + 5i)(2x - 5i)

Substituting this back into our original factorization, we get:

8x² + 50 = 2(2x + 5i)(2x - 5i)

This is the completely factored form of 8x² + 50 using complex numbers.

Applications of Factoring

The ability to factor expressions is essential in various algebraic applications. Here are a few examples:

• Solving Quadratic Equations: Factoring is a crucial technique for solving quadratic equations of the form ax² + bx + c = 0. By factoring the quadratic expression, we can find the values of x that make the equation true.
• Simplifying Expressions: Factoring simplifies complex algebraic expressions, making them easier to manipulate and understand. This is especially helpful when working with rational expressions (fractions with algebraic terms).
• Calculus: Factoring plays a vital role in calculus, particularly in differentiation and integration.
• Graphing Quadratic Functions: The factored form of a quadratic function readily reveals the x-intercepts (roots) of the function, aiding in accurately graphing the parabola.

Common Mistakes to Avoid

When factoring expressions, several common mistakes can occur:

• Forgetting the GCF: Always check for a greatest common factor before applying other factoring techniques. Failing to do so can lead to incomplete factorization.
• Incorrect Application of Factoring Techniques: Ensure you understand and apply the correct factoring technique for the given expression. Mistakes in applying difference of squares, trinomial factoring, or grouping can lead to errors.
• Incomplete Factoring: Always check if the factored expression can be further simplified or factored.

Conclusion

The completely factored form of 8x² + 50 over real numbers is 2(4x² + 25). While 4x² + 25 is not factorable over the real numbers, it can be factored using complex numbers as 2(2x + 5i)(2x - 5i). Understanding the process of factoring, mastering different factoring techniques, and avoiding common pitfalls are critical skills for success in algebra and beyond. The ability to factor expressions is a cornerstone of numerous mathematical applications, making it a fundamental skill to develop and hone. Through practice and careful attention to detail, mastering factoring becomes significantly easier and more intuitive. Remember to always check your work and ensure that your factored expression, when expanded, returns to the original expression.