Factored Form Of X^2 - 4x - 5 A Step-by-Step Guide

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In the realm of algebra, factoring quadratic expressions is a fundamental skill. Mastering this technique unlocks the ability to solve quadratic equations, simplify algebraic fractions, and delve into more advanced mathematical concepts. This comprehensive guide will walk you through the process of factoring the quadratic expression x2−4x−5x^2 - 4x - 5, providing a step-by-step explanation and ensuring you grasp the underlying principles.

Understanding Quadratic Expressions

Before we dive into the specifics of factoring x2−4x−5x^2 - 4x - 5, let's establish a solid understanding of quadratic expressions in general. A quadratic expression is a polynomial expression of degree two. The standard form of a quadratic expression is:

ax2+bx+cax^2 + bx + c

where a, b, and c are constants, and x is the variable. In our case, the expression x2−4x−5x^2 - 4x - 5 fits this form, with a = 1, b = -4, and c = -5. Factoring a quadratic expression involves rewriting it as a product of two linear expressions. This is essentially the reverse process of expanding two binomials.

The Factoring Process: Unveiling the Steps

Now, let's embark on the journey of factoring x2−4x−5x^2 - 4x - 5. The key to factoring this quadratic expression lies in finding two numbers that satisfy specific conditions. We need two numbers that:

  1. Multiply to give the constant term (c), which is -5 in our case.
  2. Add up to give the coefficient of the linear term (b), which is -4 in our case.

This is a crucial step, and it often involves a bit of trial and error. Let's systematically explore the possibilities. We need to find two numbers whose product is -5. The possible pairs of factors of -5 are:

  • -1 and 5
  • 1 and -5

Now, let's check which of these pairs adds up to -4:

  • -1 + 5 = 4
  • 1 + (-5) = -4

Aha! We've found our pair. The numbers 1 and -5 satisfy both conditions. They multiply to -5 and add up to -4. This is a critical breakthrough in the factoring process.

With these numbers in hand, we can rewrite the middle term (-4x) of the quadratic expression as the sum of the terms formed by using 1 and -5 as coefficients of x:

x2−4x−5=x2+1x−5x−5x^2 - 4x - 5 = x^2 + 1x - 5x - 5

Notice that we've simply split the -4x term into 1x and -5x. This manipulation is a key technique in factoring by grouping.

Now, we can apply the factoring by grouping method. We group the first two terms and the last two terms:

(x2+1x)+(−5x−5)(x^2 + 1x) + (-5x - 5)

Next, we factor out the greatest common factor (GCF) from each group:

x(x+1)−5(x+1)x(x + 1) - 5(x + 1)

Observe that we now have a common binomial factor of (x + 1). This is a telltale sign that we're on the right track. We factor out this common binomial factor:

(x+1)(x−5)(x + 1)(x - 5)

And there you have it! We've successfully factored the quadratic expression x2−4x−5x^2 - 4x - 5 into the product of two linear expressions: (x+1)(x−5)(x + 1)(x - 5).

Verifying the Result: A Crucial Step

To ensure our factoring is correct, it's always a good practice to verify the result. We can do this by expanding the factored form and checking if it matches the original quadratic expression. Let's expand (x+1)(x−5)(x + 1)(x - 5) using the distributive property (often referred to as the FOIL method):

(x+1)(x−5)=x(x−5)+1(x−5)(x + 1)(x - 5) = x(x - 5) + 1(x - 5)

=x2−5x+x−5= x^2 - 5x + x - 5

=x2−4x−5= x^2 - 4x - 5

As you can see, the expanded form matches the original quadratic expression, confirming that our factoring is indeed correct. This verification step provides confidence in our solution.

The Correct Answer: D. (x−5)(x+1)(x-5)(x+1)

Based on our factoring process, the factored form of x2−4x−5x^2 - 4x - 5 is (x−5)(x+1)(x - 5)(x + 1). Therefore, the correct answer is D. (x−5)(x+1)(x-5)(x+1). Note that the order of the factors doesn't matter due to the commutative property of multiplication. So, (x+1)(x−5)(x + 1)(x - 5) is equivalent to (x−5)(x+1)(x - 5)(x + 1).

Common Mistakes to Avoid

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

  1. Incorrectly Identifying the Signs: Pay close attention to the signs of the factors. A common mistake is to get the signs mixed up, leading to an incorrect factored form. Always double-check that the factors multiply to give the correct constant term and add up to give the correct coefficient of the linear term.
  2. Forgetting to Factor Out the GCF: Before attempting to factor a quadratic expression, always check if there's a greatest common factor (GCF) that can be factored out from all the terms. Factoring out the GCF simplifies the expression and makes it easier to factor the remaining quadratic.
  3. Stopping Too Early: Make sure you've factored the expression completely. Sometimes, after the first factoring step, one or both of the factors might be further factorable. Always check if the resulting factors can be factored again.
  4. Skipping the Verification Step: As we emphasized earlier, verifying your result is crucial. Expanding the factored form and comparing it to the original expression helps catch any mistakes and ensures that your factoring is correct.

Mastering Factoring: Practice Makes Perfect

Factoring quadratic expressions is a skill that improves with practice. The more you practice, the more comfortable and confident you'll become in applying the techniques. Here are some tips for mastering factoring:

  1. Start with Simple Examples: Begin with factoring simple quadratic expressions where the coefficients are small and the factoring is straightforward. This builds a solid foundation and helps you understand the basic concepts.
  2. Gradually Increase the Difficulty: As you become more proficient, tackle more challenging quadratic expressions with larger coefficients and more complex factoring patterns. This stretches your skills and prepares you for a wider range of problems.
  3. Work Through a Variety of Problems: Practice factoring different types of quadratic expressions, including those with a leading coefficient other than 1, those with perfect square trinomials, and those with differences of squares. This exposure to different problem types enhances your understanding and flexibility.
  4. Seek Help When Needed: Don't hesitate to ask for help if you're struggling with a particular problem or concept. Consult your teacher, classmates, or online resources. Collaboration and seeking clarification are essential for learning.
  5. Practice Regularly: Consistent practice is key to mastering factoring. Set aside dedicated time for practice, even if it's just for a few minutes each day. Regular practice reinforces your skills and prevents them from becoming rusty.

Beyond Factoring: Applications in Mathematics

Factoring quadratic expressions is not just an isolated skill; it's a stepping stone to more advanced mathematical concepts. Factoring plays a crucial role in:

  1. Solving Quadratic Equations: Factoring is a primary method for solving quadratic equations. By setting the factored form of a quadratic expression equal to zero, we can use the zero-product property to find the roots (solutions) of the equation.
  2. Simplifying Algebraic Fractions: Factoring is used to simplify algebraic fractions. By factoring the numerator and denominator, we can cancel out common factors and reduce the fraction to its simplest form.
  3. Graphing Quadratic Functions: Factoring helps in graphing quadratic functions. The roots of the quadratic equation (obtained by factoring) represent the x-intercepts of the parabola, which is the graph of the quadratic function.
  4. Calculus and Beyond: Factoring is a fundamental skill that is used extensively in calculus and other higher-level mathematics courses. It's essential for simplifying expressions, solving equations, and performing various mathematical operations.

Conclusion: Embracing the Power of Factoring

Factoring the quadratic expression x2−4x−5x^2 - 4x - 5 is a journey into the heart of algebraic manipulation. By understanding the underlying principles, mastering the techniques, and practicing diligently, you can unlock the power of factoring and apply it to a wide range of mathematical problems. Remember the steps: find the numbers, rewrite the middle term, factor by grouping, and verify your result. With consistent effort and a keen eye for detail, you'll become a factoring pro in no time. The factored form of x2−4x−5x^2 - 4x - 5 is indeed (x−5)(x+1)(x - 5)(x + 1), a testament to the elegance and power of algebraic factorization.