Factoring 4x² + 8x - 60 A Step-by-Step Guide

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Factoring expressions is a fundamental skill in algebra, and mastering it opens doors to solving a wide range of mathematical problems. In this comprehensive guide, we will delve into the process of factoring the quadratic expression 4x² + 8x - 60 completely. Our approach will be step-by-step, ensuring a clear understanding of each stage. We'll explore the underlying principles, employ effective techniques, and illustrate the process with detailed explanations. By the end of this guide, you'll be well-equipped to tackle similar factoring challenges with confidence.

1. Understanding the Basics of Factoring

Before we dive into the specifics of our expression, let's establish a firm grasp of the basics of factoring. At its core, factoring involves breaking down a mathematical expression into its constituent parts, or factors. These factors, when multiplied together, yield the original expression. Factoring is like reverse multiplication, allowing us to deconstruct a complex expression into simpler components. In the context of quadratic expressions, factoring involves expressing a trinomial (an expression with three terms) as the product of two binomials (expressions with two terms).

Why is factoring so important? It's a powerful tool with numerous applications in algebra and beyond. Factoring is crucial for solving quadratic equations, simplifying algebraic expressions, and analyzing mathematical functions. It's a building block for more advanced mathematical concepts, making it an essential skill for students and professionals alike. Think of factoring as a key that unlocks solutions to a variety of mathematical problems.

Consider the simple example of factoring the number 12. We can express 12 as the product of several pairs of factors, such as 1 x 12, 2 x 6, or 3 x 4. Each of these pairs represents a different way of factoring 12. Similarly, when factoring algebraic expressions, we seek to find the binomials that, when multiplied, produce the original expression. This process often involves identifying common factors, applying specific factoring patterns, and carefully arranging the terms to achieve the desired result. The more you practice factoring, the more adept you'll become at recognizing patterns and applying the appropriate techniques.

2. Step 1: Identifying the Greatest Common Factor (GCF)

When faced with a factoring problem, the first step is often to identify the greatest common factor (GCF) of all the terms in the expression. The GCF is the largest number and/or variable that divides evenly into each term. Finding the GCF simplifies the expression and makes subsequent factoring steps easier. In our expression, 4x² + 8x - 60, we need to determine the GCF of the coefficients (4, 8, and -60) and the variable terms (x² and x).

Let's start by finding the GCF of the coefficients. The factors of 4 are 1, 2, and 4. The factors of 8 are 1, 2, 4, and 8. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The largest number that appears in all three lists is 4. Therefore, the GCF of the coefficients is 4. Now, let's consider the variable terms. The term 4x² has two factors of x (x * x), and the term 8x has one factor of x. The constant term -60 has no x factor. The greatest common factor involving x is x raised to the lowest power present in the terms, which in this case is x¹ or simply x. However, since -60 has no x term, the GCF for the variable part is just 1.

Combining our findings, the overall GCF of the expression 4x² + 8x - 60 is 4. This means we can factor out a 4 from each term in the expression. Factoring out the GCF is like reverse distribution. We divide each term by the GCF and write the result in parentheses. This step reduces the complexity of the expression, paving the way for further factoring.

3. Step 2: Factoring out the GCF

Now that we've identified the GCF as 4, we can proceed to factor it out of the expression 4x² + 8x - 60. To do this, we divide each term in the expression by 4:

  • (4x²) / 4 = x²
  • (8x) / 4 = 2x
  • (-60) / 4 = -15

We now rewrite the original expression by placing the GCF outside parentheses and the results of the division inside:

4(x² + 2x - 15)

This step is crucial because it simplifies the expression inside the parentheses, making it easier to factor further. By factoring out the GCF, we've reduced the coefficients, which often makes the subsequent factoring process more manageable. Notice how the numbers inside the parentheses are smaller and easier to work with compared to the original expression. This simplification is a key benefit of factoring out the GCF.

The expression inside the parentheses, (x² + 2x - 15), is a quadratic trinomial. Our next step is to factor this trinomial into two binomials. Factoring a trinomial involves finding two binomials that, when multiplied together, produce the trinomial. This process often requires some trial and error, but there are systematic approaches we can use to make it more efficient. By factoring out the GCF first, we've effectively reduced the complexity of the trinomial, making the factoring process less daunting.

4. Step 3: Factoring the Trinomial (x² + 2x - 15)

With the GCF factored out, we now focus on factoring the trinomial (x² + 2x - 15). This trinomial is in the standard quadratic form ax² + bx + c, where a = 1, b = 2, and c = -15. To factor this trinomial, we need to find two numbers that multiply to c (-15) and add up to b (2). This is a common strategy for factoring quadratic trinomials where the leading coefficient (a) is 1.

Let's systematically identify the pairs of factors of -15:

  • 1 and -15
  • -1 and 15
  • 3 and -5
  • -3 and 5

Now, we need to determine which of these pairs adds up to 2. Let's check each pair:

  • 1 + (-15) = -14
  • -1 + 15 = 14
  • 3 + (-5) = -2
  • -3 + 5 = 2

The pair -3 and 5 satisfies our conditions: they multiply to -15 and add up to 2. This means we can rewrite the trinomial as the product of two binomials, using -3 and 5 as the constant terms in those binomials.

Therefore, the factored form of (x² + 2x - 15) is (x - 3)(x + 5). This step is the heart of factoring the trinomial. We've successfully decomposed the trinomial into two binomials that, when multiplied together, will give us the original trinomial. Now, we need to combine this result with the GCF we factored out earlier to get the complete factored form of the original expression.

5. Step 4: Combining the GCF and the Factored Trinomial

We've successfully factored the trinomial (x² + 2x - 15) into (x - 3)(x + 5). Now, we need to bring back the GCF we factored out in Step 2, which was 4. To obtain the completely factored form of the original expression 4x² + 8x - 60, we simply multiply the GCF by the factored trinomial:

4(x - 3)(x + 5)

This is the final factored form of the expression. We have successfully broken down the original expression into its constituent factors: 4, (x - 3), and (x + 5). When these factors are multiplied together, they will produce the original expression, 4x² + 8x - 60. This completes the factoring process.

It's always a good practice to check your work by multiplying the factors back together to ensure they indeed yield the original expression. This step helps to verify that your factoring is correct and that you haven't made any errors along the way. In this case, if we multiply 4, (x - 3), and (x + 5), we should obtain 4x² + 8x - 60. Let's perform the multiplication to confirm our result.

First, we'll multiply the binomials (x - 3) and (x + 5):

(x - 3)(x + 5) = x² + 5x - 3x - 15 = x² + 2x - 15

Now, we multiply the result by the GCF, 4:

4(x² + 2x - 15) = 4x² + 8x - 60

As we can see, the result matches our original expression, confirming that our factoring is correct. This check provides assurance that we have successfully factored the expression completely.

6. Conclusion: The Complete Factorization

In this guide, we have walked through the process of factoring the expression 4x² + 8x - 60 completely. We began by understanding the basics of factoring, emphasizing its importance in algebra and its applications in solving equations and simplifying expressions. We then followed a step-by-step approach:

  1. Identified the Greatest Common Factor (GCF): We determined that the GCF of the terms in the expression was 4.
  2. Factored out the GCF: We factored out the 4, resulting in the expression 4(x² + 2x - 15).
  3. Factored the Trinomial: We factored the trinomial (x² + 2x - 15) into (x - 3)(x + 5) by finding two numbers that multiply to -15 and add up to 2.
  4. Combined the GCF and the Factored Trinomial: We combined the GCF and the factored trinomial to obtain the complete factored form: 4(x - 3)(x + 5).

Therefore, the completely factored form of the expression 4x² + 8x - 60 is 4(x - 3)(x + 5). This comprehensive factorization provides a clear representation of the expression in its simplest factored form. By following these steps, you can confidently tackle similar factoring problems and expand your algebraic skills.

Factoring is a skill that improves with practice. The more you practice factoring different types of expressions, the more proficient you will become. Don't be discouraged if you encounter challenges along the way. Factoring can sometimes be tricky, but with persistence and a solid understanding of the principles, you can master it. Remember to always look for the GCF first, and then apply appropriate techniques for factoring the remaining expression. With dedication and practice, you'll become a factoring expert!