Finding Common Factors Of Polynomials A Step By Step Guide

Determining the common factors of polynomial expressions is a fundamental concept in algebra. This article provides a step-by-step solution to the question: Which expression is a factor of both $x^2 - 9$ and $x^2 + 8x + 15$? We will explore various factoring techniques and demonstrate how to identify the correct answer from the given options. Understanding factoring is crucial for solving polynomial equations, simplifying algebraic fractions, and tackling more advanced mathematical problems. Let's dive into the solution!

Understanding the Problem

Before we begin, let’s restate the problem clearly. We are given two quadratic expressions:

1. $x^2 - 9$
2. $x^2 + 8x + 15$

Our goal is to find the expression that is a factor of both quadratics. This means that both expressions can be divided evenly by this common factor. The options provided are:

A. $(x+5)$ B. $(x+3)$ C. $(x-3)$ D. $(x-9)$

To solve this, we need to factor each quadratic expression and then compare the factors to see which one appears in both.

Factoring $x^2 - 9$

The first expression, $x^2 - 9$, is a difference of squares. Recognizing patterns like the difference of squares is a key skill in factoring. The difference of squares pattern is given by:

$a^2 - b^2 = (a + b)(a - b)$

In our case, $x^2 - 9$ can be seen as $x^2 - 3^2$. Applying the difference of squares pattern, we have:

$x^2 - 9 = (x + 3)(x - 3)$

Thus, the factors of $x^2 - 9$ are $(x + 3)$ and $(x - 3)$. This is a crucial step as it immediately narrows down our potential answers to either option B or option C. Understanding and applying the difference of squares factorization is a foundational skill in algebra, applicable not only to quadratic expressions but also to more complex polynomials. This pattern allows us to break down the expression into simpler, manageable factors, making it easier to identify common factors with other expressions. Factoring is essentially the reverse process of expansion, and recognizing these patterns helps streamline the factorization process. The result $(x+3)(x-3)$ indicates that both $(x+3)$ and $(x-3)$ could potentially be the common factor we're looking for, which means we need to move on to the next expression to confirm our answer.

Factoring $x^2 + 8x + 15$

Now, let’s factor the second expression, $x^2 + 8x + 15$. This is a quadratic trinomial of the form $ax^2 + bx + c$, where $a = 1$, $b = 8$, and $c = 15$. To factor this type of expression, we need to find two numbers that multiply to $c$ (15) and add up to $b$ (8). These numbers will help us break down the middle term and rewrite the trinomial in a factored form. This method is widely used for factoring quadratic expressions and is an essential tool for simplifying algebraic expressions and solving equations. The approach involves systematically identifying number pairs that satisfy the multiplication condition and then checking if their sum matches the coefficient of the middle term. This process requires a good understanding of number properties and the ability to quickly assess the compatibility of different number combinations. Let's explore the pairs of factors for 15 to identify the correct combination.

The pairs of factors for 15 are:

• 1 and 15
• 3 and 5

Out of these pairs, 3 and 5 add up to 8, which is the coefficient of our middle term ($8x$). Therefore, we can rewrite the quadratic expression as:

$x^2 + 8x + 15 = x^2 + 3x + 5x + 15$

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

$(x^2 + 3x) + (5x + 15)$

Factor out the greatest common factor (GCF) from each group:

$x(x + 3) + 5(x + 3)$

Notice that $(x + 3)$ is a common factor in both terms. Factor it out:

$(x + 3)(x + 5)$

So, the factors of $x^2 + 8x + 15$ are $(x + 3)$ and $(x + 5)$. This result is critical because it provides us with another set of factors that we can compare with the factors of the first expression. Factoring by grouping is a versatile technique that can be applied to various polynomial expressions, especially when dealing with four or more terms. The ability to identify common factors within groups is crucial for simplifying expressions and solving equations. This method demonstrates the power of algebraic manipulation in breaking down complex expressions into more manageable components.

Identifying the Common Factor

Now that we have factored both expressions:

1. $x^2 - 9 = (x + 3)(x - 3)$
2. $x^2 + 8x + 15 = (x + 3)(x + 5)$

We can easily identify the common factor. By comparing the factors of both expressions, we see that $(x + 3)$ appears in both factorizations. This means that $(x + 3)$ is a factor of both $x^2 - 9$ and $x^2 + 8x + 15$. Identifying common factors is a crucial skill in simplifying algebraic expressions and solving equations. It allows us to reduce complex problems into simpler, more manageable forms. The process of finding common factors often involves comparing the factored forms of different expressions and identifying the elements that they share. This skill is not only useful in algebra but also in higher-level mathematics, such as calculus and differential equations. The ability to quickly and accurately identify common factors can significantly speed up problem-solving and reduce the likelihood of errors.

The Correct Answer

Therefore, the expression that is a factor of both $x^2 - 9$ and $x^2 + 8x + 15$ is $(x + 3)$. This corresponds to option B. This is a definitive answer based on the factorization and comparison process we've undertaken. The correct identification of the common factor demonstrates a clear understanding of factorization techniques and the ability to apply them effectively. Selecting the right answer from multiple choices often requires a thorough understanding of the underlying concepts and the ability to systematically eliminate incorrect options.

Why Other Options Are Incorrect

Let’s briefly discuss why the other options are incorrect:

A. $(x + 5)$: This is a factor of $x^2 + 8x + 15$ but not of $x^2 - 9$. B. $(x + 3)$: This is the correct answer, as we have shown. C. $(x - 3)$: This is a factor of $x^2 - 9$ but not of $x^2 + 8x + 15$. D. $(x - 9)$: This is not a factor of either expression.

Understanding why incorrect options are wrong is as important as knowing the correct answer. It reinforces the concepts and helps in avoiding similar mistakes in the future. This process involves carefully examining each option and comparing it with the results of our factorization. By eliminating incorrect options, we can build confidence in our understanding of the problem and the solution we have arrived at. This step-by-step analysis also helps in developing critical thinking skills, which are essential for success in mathematics and beyond.

Conclusion

In conclusion, by factoring both quadratic expressions, $x^2 - 9$ and $x^2 + 8x + 15$, we found that the common factor is $(x + 3)$. This makes option B the correct answer. Mastering factoring techniques and recognizing patterns like the difference of squares is essential for success in algebra. Factoring is a fundamental skill that opens doors to solving complex equations and understanding polynomial behavior. The ability to factor expressions quickly and accurately is a key asset in mathematical problem-solving.

This problem highlights the importance of a systematic approach to solving mathematical questions. By breaking down the problem into smaller, manageable steps, we were able to identify the correct solution efficiently. This step-by-step approach is a valuable strategy for tackling challenging problems in various fields. Consistent practice and a solid understanding of fundamental concepts are crucial for building confidence and competence in mathematics.