Factoring Quadratics: Finding Factors Of X² + 6x + 8

Hey math enthusiasts! Let's dive into the fascinating world of factoring quadratics. We're going to explore a problem where we're given a cubic function, find one of its roots, and then delve into factoring a resulting quadratic expression. This is a common type of problem in algebra, and understanding the steps involved is key to mastering this area of mathematics. Ready to break down some equations and understand their core components? Let's get started!

The Problem Unpacked: A Step-by-Step Guide

Our journey begins with the cubic function: $f(x) = x^3 + x^2 - 22x - 40$. We're told that one of the roots of this function is $5$. Remember, a root of a function is a value of x that makes the function equal to zero (i.e., $f(x) = 0$). This means if we plug in x = 5, the entire expression will evaluate to zero. It's like finding a special key (the root) that unlocks the equation's solution!

Next, we're informed that dividing the cubic expression $(x^3 + x^2 - 22x - 40)$ by $(x - 5)$ results in $x^2 + 6x + 8$. This division process is a crucial step in simplifying the problem and bringing us closer to our goal. When we divide a polynomial by one of its factors, we're essentially 'undoing' the multiplication that created the original expression. In this case, since $(x-5)$ is a factor of the cubic equation, then we know $(x-5)$ will evenly divide into the cubic equation. This step is a powerful way to break down complex expressions into simpler forms. We can represent the relationship as follows: $(x^3 + x^2 - 22x - 40) = (x - 5)(x^2 + 6x + 8)$.

Finally, the question asks us to identify the factor(s) of the quadratic expression $x^2 + 6x + 8$. This is where our factoring skills come into play. We need to find two binomials (expressions with two terms) that, when multiplied together, give us $x^2 + 6x + 8$. Essentially, we are looking for two numbers that sum to 6 (the coefficient of the x term) and multiply to 8 (the constant term). Factoring is a fundamental skill in algebra, enabling you to solve quadratic equations, simplify expressions, and understand the behavior of functions. Finding these factors will allow us to rewrite the quadratic equation in a more useful form, revealing its roots and providing deeper insights into its characteristics.

Now, let's explore this step in more detail, understanding how to factor the quadratic $x^2 + 6x + 8$.

Factoring the Quadratic: $x^2 + 6x + 8$

Now, let's focus on the quadratic expression $x^2 + 6x + 8$. Our objective here is to factor this quadratic into two binomial expressions. Factoring is the reverse process of expanding (or multiplying) binomials. It is essentially breaking down a quadratic expression into its constituent parts, which can provide invaluable information about the quadratic, like its roots. Remember, the general form of a quadratic is $ax^2 + bx + c$, where a, b, and c are constants. In our example, a = 1, b = 6, and c = 8.

To factor a quadratic of the form $x^2 + bx + c$, we need to find two numbers that: 1) Multiply to give us c (the constant term, which is 8 in our case) and 2) Add up to give us b (the coefficient of the x term, which is 6 in our case).

Let's brainstorm the factor pairs of 8: (1, 8) and (2, 4). Now, let's examine each pair to see which one adds up to 6. 1 + 8 = 9 (not 6) and 2 + 4 = 6 (Bingo!). The numbers 2 and 4 satisfy both conditions. Now that we've identified the numbers, we can write the factored form of the quadratic.

Since our numbers are 2 and 4, we can write the factored form as $(x + 2)(x + 4)$. This means that $(x + 2)$ and $(x + 4)$ are the factors of $x^2 + 6x + 8$. To verify, we can expand our factors by using the FOIL method (First, Outer, Inner, Last): $(x+2)(x+4) = x*x + x*4 + 2*x + 2*4 = x^2 + 4x + 2x + 8 = x^2 + 6x + 8$.

Therefore, we've successfully factored the quadratic expression. From this, we know that two of the factors are $(x+2)$ and $(x+4)$. In the context of the original cubic equation, these factors, along with (x - 5) will allow you to find the roots of the cubic equation. Understanding this process enhances problem-solving skills and is the foundation for more advanced topics in algebra and calculus.

Matching Factors to Answer Choices

Given the options, let's look for our factors: $(x+2)$ and $(x+4)$:

• A. $(x - 4)$ - This is incorrect, as our factor is $(x + 4)$.
• B. $(x + 4)$ - This is correct, as we found this factor in our calculations.
• C. $(x - 2)$ - This is incorrect, as our factor is $(x + 2)$.
• D. $(x + 2)$ - This is correct, as we found this factor in our calculations.

So, the correct answers are B and D. Understanding these components of factoring will help you simplify complex mathematical problems and build a strong foundation for future mathematical studies. By recognizing the patterns, you will enhance your problem-solving abilities and become more confident in navigating mathematical challenges.

Mastering Quadratics and Beyond

Mastering quadratic equations and factoring is not just about solving problems; it's about developing critical thinking and problem-solving skills that are essential in various fields. From engineering and physics to computer science and economics, a solid understanding of quadratics is a cornerstone for advanced studies. Furthermore, the ability to break down complex problems into smaller, manageable parts (as we did with factoring) is a valuable skill in all aspects of life.

As you continue to practice, you'll start to recognize patterns and become more comfortable with different types of quadratic equations. Remember, the key is to understand the underlying principles. Practice makes perfect, so don't be discouraged if it takes some time to grasp these concepts. Each problem you solve is a step towards greater understanding and mathematical proficiency.

Continue to work through similar problems, exploring different scenarios and variations. This will help you deepen your understanding and gain confidence in your ability to solve quadratic equations and tackle more complex mathematical problems. By doing so, you'll develop a stronger foundation for success in mathematics and beyond.

In summary: We started with a cubic function, used a root to simplify it, and then factored the resulting quadratic. We identified that the factors of $x^2 + 6x + 8$ are $(x + 2)$ and $(x + 4)$. This process demonstrates the power of factoring and its application in solving polynomial equations.

To further your understanding, consider exploring more complex factoring techniques, such as factoring by grouping and using the quadratic formula, to solve more complicated equations. Keep practicing, and you'll be well on your way to mastering quadratic equations! Your journey into the world of mathematics is just beginning; embrace the challenges and enjoy the process of learning.

Here are some related websites that you might find helpful:

• Khan Academy: (https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics) - A great resource for video tutorials and practice exercises on quadratic equations and factoring.
• Math is Fun: (https://www.mathsisfun.com/algebra/factoring-quadratics.html) - Provides clear explanations and examples on factoring quadratics.

Happy learning! Keep exploring the wonderful world of mathematics! These resources should offer additional learning opportunities to strengthen your comprehension and provide more practice. Good luck, and keep up the great work!