Factorization Of 3x² - 10x + 8 A Step-by-Step Guide
In the realm of algebra, factorization stands as a cornerstone skill, enabling us to break down complex expressions into simpler, manageable components. Mastering this technique unlocks a deeper understanding of polynomial behavior, simplifies equation solving, and paves the way for advanced mathematical concepts. This comprehensive guide delves into the factorization of the quadratic expression 3x² - 10x + 8, providing a step-by-step approach to unraveling its factors and illuminating the underlying principles. Let's embark on this journey of algebraic exploration, equipping ourselves with the tools and knowledge to conquer quadratic factorization with confidence.
Understanding the Basics of Factorization
Before we dive into the specifics of 3x² - 10x + 8, let's lay a solid foundation by revisiting the core concepts of factorization. In essence, factorization is the reverse process of expansion. While expansion involves multiplying out expressions to obtain a simplified form, factorization involves dissecting an expression into its constituent factors – expressions that, when multiplied together, yield the original expression. Think of it like reverse engineering; we're taking the final product and figuring out the ingredients that went into making it.
For instance, consider the number 12. We can factorize it as 2 × 6, 3 × 4, or even 2 × 2 × 3. Each of these pairs (or sets) of numbers represents a factorization of 12. Similarly, in algebra, we can factorize expressions involving variables, such as polynomials. Polynomial factorization plays a pivotal role in solving equations, simplifying expressions, and analyzing the behavior of functions.
One particularly important type of expression to factorize is the quadratic expression, which takes the general form ax² + bx + c, where a, b, and c are constants. Our target expression, 3x² - 10x + 8, falls squarely into this category. To factorize a quadratic expression, we aim to rewrite it as a product of two linear expressions – expressions of the form (px + q) and (rx + s), where p, q, r, and s are constants. The challenge lies in finding the correct combination of constants that satisfy the factorization. The factorization process can sometimes seem like solving a puzzle, where we need to find the right pieces that fit together to form the whole picture.
Deconstructing 3x² - 10x + 8: A Step-by-Step Approach
Now, let's turn our attention to the heart of the matter: factorizing the expression 3x² - 10x + 8. We'll employ a systematic approach that involves identifying key coefficients, finding the right factors, and piecing them together to form the final factorization. The method we'll use here is often referred to as the "splitting the middle term" method, a powerful technique for factorizing quadratic expressions.
Step 1: Identifying the Coefficients
The first step in our factorization journey is to identify the coefficients of the quadratic expression. In 3x² - 10x + 8, we have:
- a = 3 (the coefficient of x²)
- b = -10 (the coefficient of x)
- c = 8 (the constant term)
These coefficients hold the key to unlocking the factorization. They guide us in finding the numbers that will allow us to split the middle term and rewrite the expression in a way that facilitates factorization. Understanding these coefficients is like having the combination to a lock; it's the first step toward opening up the solution.
Step 2: Finding the Magic Numbers
The next crucial step is to find two numbers that satisfy two specific conditions:
- Their product equals the product of a and c (i.e., a × c = 3 × 8 = 24).
- Their sum equals b (i.e., -10).
This is where the puzzle-solving aspect of factorization comes into play. We need to brainstorm pairs of numbers whose product is 24 and whose sum is -10. Let's consider the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Since we need a negative sum, we'll consider negative factors as well. After some deliberation, we'll find that the numbers -4 and -6 satisfy both conditions:
- (-4) × (-6) = 24
- (-4) + (-6) = -10
These numbers are our "magic numbers." They are the key to splitting the middle term and proceeding with factorization. Finding these numbers might require some trial and error, but with practice, you'll develop an intuition for identifying them quickly.
Step 3: Splitting the Middle Term
Now that we've found our magic numbers, we can split the middle term (-10x) using these numbers. We rewrite the original expression as:
3x² - 10x + 8 = 3x² - 4x - 6x + 8
Notice that we've replaced -10x with -4x - 6x. This seemingly simple step is the cornerstone of the splitting the middle term method. By splitting the middle term in this way, we create opportunities for grouping and factoring in the next step.
Step 4: Factoring by Grouping
With the middle term split, we can now factorize by grouping. We group the first two terms and the last two terms together:
(3x² - 4x) + (-6x + 8)
Next, we factorize out the greatest common factor (GCF) from each group. From the first group (3x² - 4x), the GCF is x. From the second group (-6x + 8), the GCF is -2. Factoring these out, we get:
x(3x - 4) - 2(3x - 4)
Notice that we now have a common factor of (3x - 4) in both terms. This is a crucial sign that we're on the right track. If we don't have a common factor at this stage, it indicates an error in our previous steps, and we need to revisit our calculations.
Step 5: Final Factorization
Finally, we can factorize out the common factor (3x - 4) from the entire expression:
(3x - 4)(x - 2)
This is the complete factorization of 3x² - 10x + 8. We've successfully broken down the quadratic expression into the product of two linear factors.
Verifying the Factorization
To ensure the accuracy of our factorization, it's always a good practice to verify our result. We can do this by expanding the factored expression and checking if it matches the original expression. Let's expand (3x - 4)(x - 2):
(3x - 4)(x - 2) = 3x(x - 2) - 4(x - 2)
= 3x² - 6x - 4x + 8
= 3x² - 10x + 8
The expanded expression matches our original expression, confirming that our factorization is correct. This verification step provides a sense of assurance and reinforces our understanding of the factorization process.
Addressing the Question: Completing the Factorization
The original question posed a slightly different challenge: Complete the factorization of 3x² - 10x + 8, given that one factor is (x - 2). In other words, it presented us with a partially factored expression:
3x² - 10x + 8 = (x - 2)(_x - 4)
Our task is to determine the missing coefficient in the second factor. We've already factorized the expression completely, so we know that the complete factorization is:
3x² - 10x + 8 = (x - 2)(3x - 4)
Comparing this with the given partially factored expression, we can clearly see that the missing coefficient is 3. Therefore, the completed factorization is:
3x² - 10x + 8 = (x - 2)(3x - 4)
Conclusion: Mastering Quadratic Factorization
Factorizing quadratic expressions is a fundamental skill in algebra, with applications spanning various mathematical domains. In this comprehensive guide, we've delved into the factorization of 3x² - 10x + 8, demonstrating a step-by-step approach using the splitting the middle term method. We've seen how identifying coefficients, finding magic numbers, splitting the middle term, and factoring by grouping can lead us to the complete factorization of a quadratic expression.
By mastering these techniques, you'll not only be able to factorize quadratic expressions with confidence but also gain a deeper understanding of polynomial behavior and algebraic manipulation. Remember, practice is key to proficiency in factorization. The more you work through examples and apply these methods, the more comfortable and adept you'll become at factorizing quadratic expressions. Embrace the challenge, and unlock the power of factorization in your mathematical journey. Factorization is a key concept in mathematics, and it's essential to master it. Good luck!
