Factor The Polynomial $x^2 - 8x + 15$.

Factoring polynomials is a fundamental skill in algebra, and mastering it opens doors to solving various mathematical problems. In this comprehensive guide, we will delve into the process of factoring the quadratic polynomial $x^2 - 8x + 15$. We will explore different approaches, provide step-by-step explanations, and offer valuable insights to help you grasp the concept effectively. Whether you're a student looking to improve your algebra skills or simply curious about polynomial factorization, this article is designed to equip you with the knowledge and confidence you need.

Understanding Polynomial Factorization

Before we dive into the specifics of factoring $x^2 - 8x + 15$, let's first establish a solid understanding of what polynomial factorization entails. In essence, factoring a polynomial means expressing it as a product of two or more simpler polynomials. Think of it as the reverse process of polynomial multiplication. For instance, if we multiply $(x - 3)$ and $(x - 5)$, we get $x^2 - 8x + 15$. Therefore, factoring $x^2 - 8x + 15$ involves finding those two binomials that multiply together to give us the original polynomial.

Why is factoring important? Factoring polynomials is crucial because it simplifies complex expressions, making them easier to analyze and solve. It's a key technique used in solving quadratic equations, simplifying rational expressions, and even in calculus. By mastering factoring, you'll gain a powerful tool for tackling a wide range of mathematical challenges.

Different Types of Factoring: There are several methods for factoring polynomials, each suited to different types of expressions. Some common techniques include:

• Greatest Common Factor (GCF): Identifying and factoring out the largest factor common to all terms in the polynomial.
• Factoring by Grouping: Grouping terms in the polynomial and factoring out common factors from each group.
• Factoring Trinomials: Factoring quadratic expressions of the form $ax^2 + bx + c$.
• Difference of Squares: Factoring expressions in the form $a^2 - b^2$.
• Sum and Difference of Cubes: Factoring expressions in the form $a^3 + b^3$ or $a^3 - b^3$.

In this article, we will primarily focus on factoring the trinomial $x^2 - 8x + 15$, which falls under the category of factoring trinomials. We'll break down the process into manageable steps, ensuring you understand the logic behind each one.

Factoring $x^2 - 8x + 15$: A Step-by-Step Approach

Now, let's get down to the business of factoring the given polynomial, $x^2 - 8x + 15$. This is a quadratic trinomial, which means it has three terms and the highest power of the variable x is 2. Factoring such trinomials often involves finding two binomials that multiply to give the original trinomial. Here’s a step-by-step guide:

Step 1: Identify the coefficients. The first step in factoring $x^2 - 8x + 15$ is to identify the coefficients of each term. In this case, we have:

• The coefficient of $x^2$ is 1.
• The coefficient of $x$ is -8.
• The constant term is 15.

These coefficients will play a crucial role in finding the factors. Understanding the signs of these coefficients is particularly important. The positive constant term (15) indicates that the signs in the binomial factors will either be both positive or both negative. The negative coefficient of the x term (-8) tells us that both signs in the factors will be negative. This is because two negative numbers multiplied together give a positive number (the constant term), and their sum is a negative number (the coefficient of the x term).

Step 2: Find two numbers that multiply to the constant term and add up to the coefficient of the x term. This is the core of factoring trinomials. We need to find two numbers that satisfy two conditions simultaneously:

1. Their product must equal the constant term (15).
2. Their sum must equal the coefficient of the x term (-8).

Let's list the pairs of factors of 15:

• 1 and 15
• 3 and 5

Since we know that both signs must be negative, we consider the negative counterparts:

• -1 and -15
• -3 and -5

Now, let's check which pair adds up to -8:

• -1 + (-15) = -16
• -3 + (-5) = -8

We've found our numbers! -3 and -5 satisfy both conditions. They multiply to 15 and add up to -8.

Step 3: Write the factored form. Once we've found the two numbers, we can write the factored form of the trinomial. The numbers we found (-3 and -5) will be the constants in our binomial factors. The factored form is:

$x^2 - 8x + 15 = (x - 3)(x - 5)$

This means that if we multiply $(x - 3)$ and $(x - 5)$, we will get back our original trinomial, $x^2 - 8x + 15$.

Step 4: Verify the factored form. It's always a good idea to verify your factored form to ensure you haven't made any mistakes. We can do this by multiplying the binomials $(x - 3)$ and $(x - 5)$ using the distributive property (also known as the FOIL method):

$(x - 3)(x - 5) = x(x - 5) - 3(x - 5)$

$= x^2 - 5x - 3x + 15$

$= x^2 - 8x + 15$

As we can see, multiplying the factors gives us back the original trinomial, so our factored form is correct.

Common Mistakes to Avoid When Factoring

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

• Incorrect Signs: One of the most frequent errors is getting the signs wrong in the binomial factors. Always pay close attention to the signs of the coefficients in the original trinomial. Remember, a positive constant term with a negative coefficient for the x term indicates that both signs in the factors will be negative.
• Incorrect Factors: Another common mistake is choosing the wrong factors of the constant term. Make sure the factors you choose not only multiply to the constant term but also add up to the coefficient of the x term. It's helpful to list out all the possible pairs of factors before making your selection.
• Forgetting to Factor Out the GCF: Before attempting to factor a trinomial, always check if there's a greatest common factor (GCF) that can be factored out. Factoring out the GCF first can simplify the trinomial and make it easier to factor further.
• Stopping Too Early: Once you've factored a polynomial, make sure to check if the factors themselves can be factored further. For example, if one of the factors is another trinomial, you may need to factor it as well.
• Not Verifying the Answer: It's always a good practice to verify your factored form by multiplying the factors back together. This will help you catch any mistakes and ensure your answer is correct.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in factoring polynomials.

Practice Problems and Further Exploration

To solidify your understanding of factoring polynomials, it's essential to practice. Here are a few practice problems you can try:

1. Factor $x^2 + 7x + 12$
2. Factor $x^2 - 10x + 24$
3. Factor $x^2 + 5x - 14$

Try solving these problems using the step-by-step approach we discussed earlier. Remember to identify the coefficients, find the factors, write the factored form, and verify your answer. The more you practice, the more comfortable you'll become with factoring.

Further Exploration: If you're interested in learning more about factoring polynomials, there are many resources available online and in textbooks. You can explore different factoring techniques, such as factoring by grouping, the difference of squares, and the sum and difference of cubes. You can also delve into more complex polynomials and learn how to factor them. Websites like Khan Academy and textbooks on algebra are excellent resources for further learning.

Conclusion: Mastering Polynomial Factorization

In this comprehensive guide, we've explored the process of factoring the polynomial $x^2 - 8x + 15$. We've broken down the steps involved, discussed common mistakes to avoid, and provided practice problems to help you solidify your understanding. Factoring polynomials is a crucial skill in algebra, and mastering it will open doors to solving a wide range of mathematical problems.

Remember, the key to success in factoring is practice. The more you practice, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and don't be afraid to ask for help when you need it. With dedication and effort, you can master the art of polynomial factorization and unlock new levels of mathematical understanding.

The factored form of $x^2 - 8x + 15$ is $(x - 3)(x - 5)$. This factorization allows us to rewrite the quadratic trinomial as a product of two binomials, simplifying its structure and making it easier to analyze and use in various mathematical contexts.