Factoring $x^2 - 17x + 72$: A Step-by-Step Guide

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Factoring the Polynomial $x^2 - 17x + 72$: A Step-by-Step Guide

Hey guys! Let's dive into factoring the polynomial x2−17x+72x^2 - 17x + 72. Factoring polynomials is a crucial skill in algebra, and it's super useful for solving equations and simplifying expressions. We'll break down the process step by step, making it easy to understand. This guide will provide a comprehensive explanation on how to factor this quadratic expression, ensuring you grasp the underlying concepts and can apply them to similar problems. So, let's get started and master this factoring technique!

Understanding the Basics of Factoring

Before we jump into this specific problem, let's quickly recap what factoring means. Factoring a polynomial is like reverse multiplication. Think of it as breaking down a number into its prime factors, but with algebraic expressions. When we factor a polynomial, we're essentially trying to find two (or more) expressions that, when multiplied together, give us the original polynomial. This process involves identifying common factors, recognizing patterns, and applying different techniques to simplify the expression.

Factoring is a fundamental concept in algebra, serving as the bedrock for various mathematical operations such as solving equations, simplifying expressions, and analyzing functions. To truly grasp the art of factoring, it's essential to have a solid understanding of its basic principles and techniques. When we factor a polynomial, we are essentially reversing the process of multiplication. Imagine taking a polynomial and breaking it down into smaller components, each of which, when multiplied together, recreates the original polynomial. This process allows us to rewrite complex expressions into a more manageable form, making them easier to work with.

At its core, factoring involves identifying common elements within the polynomial and strategically extracting them to simplify the expression. These common elements can range from simple numerical factors to more complex algebraic terms. By pinpointing these commonalities, we can effectively reduce the polynomial into a product of simpler factors, offering a more insightful representation of its structure. In addition to identifying common factors, recognizing specific patterns within the polynomial can also significantly aid in the factoring process. Certain polynomials exhibit recognizable forms, such as the difference of squares, perfect square trinomials, or sums/differences of cubes. When we encounter these patterns, we can apply predetermined factoring formulas to expedite the process and arrive at the factored form more efficiently. Understanding these formulas and their applications is paramount in mastering the art of factoring.

Step 1: Identify the Coefficients

Okay, so we have x2−17x+72x^2 - 17x + 72. First, let's identify the coefficients. In this quadratic expression, we have:

  • The coefficient of x2x^2 is 1.
  • The coefficient of xx is -17.
  • The constant term is 72.

These coefficients are crucial because they guide our factoring process. We need to find two numbers that, when multiplied, give us the constant term (72) and, when added, give us the coefficient of the xx term (-17). This initial step is crucial as it sets the stage for the subsequent factoring process. By identifying the coefficients and constant term, we gain valuable insights into the structure of the quadratic expression, which ultimately guides our approach to factoring it. The coefficient of the x2x^2 term provides information about the leading term of the quadratic, while the coefficient of the xx term and the constant term dictate the relationships between the factors we seek. Understanding these relationships is paramount in effectively factoring the expression.

In this specific case, the coefficient of the x2x^2 term is 1, indicating a simple quadratic expression. The coefficient of the xx term, -17, tells us that the sum of the factors we're looking for should be -17, while the constant term, 72, informs us that the product of these factors should be 72. These two pieces of information serve as the foundation for our factoring strategy. By systematically analyzing these coefficients and their relationships, we can narrow down the possibilities for the factors and efficiently determine the correct factorization of the quadratic expression. This methodical approach ensures that we don't overlook any potential factors and that we arrive at the accurate factored form.

Step 2: Find Two Numbers That Multiply to 72 and Add to -17

This is the heart of factoring these types of quadratics. We need to think of two numbers that multiply to 72 and add up to -17. Since the product is positive and the sum is negative, we know that both numbers must be negative. Let's list the factor pairs of 72:

  • 1 and 72
  • 2 and 36
  • 3 and 24
  • 4 and 18
  • 6 and 12
  • 8 and 9

Now, let's consider the negative counterparts:

  • -1 and -72
  • -2 and -36
  • -3 and -24
  • -4 and -18
  • -6 and -12
  • -8 and -9

Looking at these pairs, we can see that -8 and -9 fit the bill. -8 multiplied by -9 is 72, and -8 plus -9 is -17. This step requires a systematic approach to ensure no potential factor pairs are overlooked. Listing out the factor pairs, both positive and negative, helps in identifying the combination that satisfies both conditions: a product equal to the constant term and a sum equal to the coefficient of the xx term. It's like a puzzle where we need to find the perfect fit.

For our specific polynomial, x2−17x+72x^2 - 17x + 72, we're hunting for two numbers that not only multiply to 72 but also add up to -17. The fact that the constant term is positive while the coefficient of the xx term is negative gives us a crucial clue: both numbers must be negative. This insight significantly narrows down our search, allowing us to focus solely on negative factor pairs of 72. By systematically examining these negative pairs, such as -1 and -72, -2 and -36, and so on, we can quickly identify the pair that sums up to -17. In this case, it's the dynamic duo of -8 and -9 that perfectly fits the criteria. This strategic approach to identifying factors is the cornerstone of efficient polynomial factoring.

Step 3: Write the Factored Form

Now that we've found our numbers, -8 and -9, we can write the factored form of the polynomial. The factored form is:

(x−8)(x−9)(x - 8)(x - 9)

This means that when we multiply (x−8)(x - 8) by (x−9)(x - 9), we should get back our original polynomial, x2−17x+72x^2 - 17x + 72. To verify this, let's expand the factored form. Writing the factored form is the culmination of our efforts in identifying the correct factors. Once we've found the numbers that satisfy the multiplication and addition conditions, expressing the polynomial in its factored form becomes a straightforward process. It's like putting the final piece of a puzzle into place, completing the overall picture.

In our case, having determined that -8 and -9 are the magic numbers, we can confidently write the factored form as (x−8)(x−9)(x - 8)(x - 9). This factored expression represents the original polynomial, x2−17x+72x^2 - 17x + 72, in a more concise and manageable form. Each factor, (x−8)(x - 8) and (x−9)(x - 9), represents a linear expression that, when multiplied together, yields the quadratic polynomial we started with. This transformation from a quadratic expression to a product of linear factors is the essence of factoring, and it opens up a myriad of possibilities for solving equations, simplifying expressions, and analyzing functions.

Step 4: Verify by Expanding (Optional but Recommended)

Let's expand (x−8)(x−9)(x - 8)(x - 9) to make sure we factored it correctly:

(x−8)(x−9)=x(x−9)−8(x−9)(x - 8)(x - 9) = x(x - 9) - 8(x - 9)

=x2−9x−8x+72= x^2 - 9x - 8x + 72

=x2−17x+72= x^2 - 17x + 72

Yep, it matches our original polynomial! This step is optional, but it's a great way to double-check your work and ensure you haven't made any mistakes. Expanding the factored form essentially reverses the factoring process, allowing us to confirm that we've arrived at the correct result. It's like retracing our steps to ensure we haven't veered off course.

For our polynomial, x2−17x+72x^2 - 17x + 72, we can verify our factored form, (x−8)(x−9)(x - 8)(x - 9), by meticulously expanding it. We begin by distributing xx over (x−9)(x - 9) and then -8 over (x−9)(x - 9), resulting in the expression x(x−9)−8(x−9)x(x - 9) - 8(x - 9). Subsequently, we perform the multiplications, obtaining x2−9x−8x+72x^2 - 9x - 8x + 72. Finally, we combine like terms, specifically the −9x-9x and −8x-8x terms, which yields x2−17x+72x^2 - 17x + 72. This result perfectly matches our original polynomial, confirming the accuracy of our factoring process and providing us with the confidence that we've indeed cracked the code to factoring this expression.

Answer

So, the factored form of x2−17x+72x^2 - 17x + 72 is:

x2−17x+72=(x−8)(x−9)x^2 - 17x + 72 = (x - 8)(x - 9)

And that's it! We've successfully factored the polynomial. Remember, the key is to find two numbers that multiply to the constant term and add up to the coefficient of the x term. With practice, you'll become a factoring pro!

Common Factoring Mistakes to Avoid

Factoring polynomials can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  1. Incorrect Sign Combinations: This is a frequent area of error. When looking for factors, ensure you're considering the correct signs. For instance, if the constant term is positive and the middle term is negative, both factors should be negative.
  2. Missing the Greatest Common Factor (GCF): Before diving into more complex factoring techniques, always check if there's a GCF that can be factored out. Overlooking this step can lead to more complicated factoring later on.
  3. Not Verifying the Solution: As we discussed, expanding the factored form back to the original polynomial is a crucial step. It helps catch any errors in your factoring process.
  4. Rushing Through the Process: Factoring requires careful attention to detail. Rushing can lead to overlooking potential factor pairs or making arithmetic errors. Take your time and double-check your work.

Avoiding these common mistakes can significantly improve your accuracy and efficiency in factoring polynomials. Remember, practice makes perfect, so keep at it, and you'll become more confident and proficient in no time.

Practice Problems

To solidify your understanding, try factoring these polynomials:

  1. x2−13x+42x^2 - 13x + 42
  2. x2+10x+24x^2 + 10x + 24
  3. x2−5x−14x^2 - 5x - 14

Work through these problems, applying the steps we've discussed. Check your answers by expanding the factored forms. The more you practice, the more comfortable you'll become with factoring.

Conclusion

Factoring the polynomial x2−17x+72x^2 - 17x + 72 involves finding two numbers that multiply to 72 and add to -17. By systematically listing factor pairs and considering the signs, we identified -8 and -9 as the correct numbers. This allowed us to write the factored form as (x−8)(x−9)(x - 8)(x - 9). Remember to always verify your solution by expanding the factored form to ensure it matches the original polynomial. Happy factoring, and keep practicing!