Factor The Polynomial Expression Completely: (x² - 5x + 4)(x² - 9).

In the realm of mathematics, specifically algebra, factoring polynomials is a fundamental skill. This article provides a comprehensive guide on how to completely factor the expression (x² - 5x + 4)(x² - 9). We will break down the process step-by-step, ensuring a clear understanding for both beginners and those looking to refresh their knowledge. The importance of factoring polynomials lies in its applications across various mathematical domains, including solving equations, simplifying expressions, and analyzing functions. Mastering this skill opens doors to more advanced mathematical concepts and problem-solving techniques. Before diving into the specifics of the given expression, let's establish a solid foundation by revisiting the basic principles of factoring. Factoring is essentially the reverse process of expansion. When we expand, we multiply terms together to obtain a polynomial. Factoring, on the other hand, involves breaking down a polynomial into its constituent factors, which are expressions that, when multiplied together, yield the original polynomial. There are several techniques for factoring polynomials, including finding the greatest common factor (GCF), using the difference of squares formula, recognizing perfect square trinomials, and employing the method of factoring by grouping. Each of these techniques serves a specific purpose and is applicable to different types of polynomials. In this article, we will utilize a combination of these techniques to completely factor the given expression. Factoring is not merely a mathematical exercise; it is a critical tool for simplifying complex expressions and solving equations. A well-factored polynomial can reveal its roots (the values that make the polynomial equal to zero) and simplify the process of graphing the corresponding function. In many real-world applications, mathematical models involve polynomials, and the ability to factor these polynomials allows us to analyze and interpret the models effectively. From engineering to economics, the applications of factoring are vast and varied. Therefore, a thorough understanding of factoring techniques is essential for anyone pursuing studies or careers in STEM fields.

Step-by-Step Factoring of (x² - 5x + 4)(x² - 9)

To completely factor the expression (x² - 5x + 4)(x² - 9), we'll tackle each quadratic expression individually and then combine the results. Our step-by-step factoring approach ensures clarity and accuracy in the process. We start by examining the first quadratic expression, x² - 5x + 4. This is a trinomial, and we need to find two numbers that multiply to the constant term (4) and add up to the coefficient of the linear term (-5). After some consideration, we can identify -1 and -4 as the two numbers that satisfy these conditions. Specifically, (-1) * (-4) = 4 and (-1) + (-4) = -5. This allows us to rewrite the trinomial in factored form as (x - 1)(x - 4). The process of factoring trinomials involves identifying the correct pair of numbers that satisfy both the multiplication and addition conditions. This might require some trial and error, but with practice, it becomes more intuitive. Once we've found the correct numbers, we can easily express the trinomial as a product of two binomials. Now, let's move on to the second quadratic expression, x² - 9. This expression is a classic example of the difference of squares. The difference of squares formula states that a² - b² can be factored as (a + b)(a - b). In this case, x² - 9 can be seen as x² - 3², where a = x and b = 3. Applying the difference of squares formula, we can factor x² - 9 as (x + 3)(x - 3). The difference of squares is a powerful pattern to recognize, as it provides a quick and straightforward way to factor certain types of expressions. Recognizing this pattern can significantly simplify the factoring process. With both quadratic expressions factored, we can now combine the results to obtain the complete factorization of the original expression. Substituting the factored forms of x² - 5x + 4 and x² - 9, we get (x - 1)(x - 4)(x + 3)(x - 3). This is the completely factored form of the given expression. Factoring often involves a combination of different techniques, and it's essential to be familiar with these techniques to handle a variety of expressions. By breaking down the expression into smaller parts and applying the appropriate factoring methods, we can systematically arrive at the complete factorization.

Combining the Factors for the Final Solution

After factoring each quadratic expression individually, the next step is to combine the factors to arrive at the final solution. We've successfully factored x² - 5x + 4 into (x - 1)(x - 4) and x² - 9 into (x + 3)(x - 3). Now, we simply multiply these factors together to obtain the completely factored form of the original expression. The completely factored form of (x² - 5x + 4)(x² - 9) is (x - 1)(x - 4)(x + 3)(x - 3). This represents the expression as a product of four linear factors. Each factor corresponds to a root of the polynomial, which is a value of x that makes the polynomial equal to zero. In this case, the roots are x = 1, x = 4, x = -3, and x = 3. Identifying the roots of a polynomial is one of the primary applications of factoring. The roots provide valuable information about the behavior of the polynomial function, such as where it crosses the x-axis on a graph. The factored form of a polynomial also makes it easier to analyze its behavior. For example, we can quickly determine the sign of the polynomial in different intervals by examining the signs of each factor. This information can be used to sketch the graph of the polynomial function or to solve inequalities involving the polynomial. In addition to finding roots, the factored form of a polynomial can be used to simplify algebraic fractions. If the numerator and denominator of a fraction share a common factor, we can cancel that factor to reduce the fraction to its simplest form. Factoring is an essential tool in this process. Furthermore, factoring is used in calculus to find limits, derivatives, and integrals of polynomial functions. The factored form often simplifies these calculations, making them more manageable. Understanding how to combine factors to achieve the final solution is crucial in mastering polynomial factorization. It represents the culmination of the individual factoring steps and provides a complete representation of the original expression in its simplest form. The ability to factor completely is a fundamental skill that has wide-ranging applications in mathematics and related fields.

Additional Examples and Practice

To solidify your understanding of factoring polynomials, let's explore additional examples and practice problems. This will help you develop the necessary skills and confidence to tackle a variety of factoring challenges. Consider the expression (6x² + 9x)(x² - 4x + 4). This expression presents a slightly different challenge compared to the previous example, as it involves factoring out a common factor and recognizing a perfect square trinomial. First, let's focus on the expression 6x² + 9x. We can identify a common factor of 3x in both terms. Factoring out 3x, we get 3x(2x + 3). Factoring out the greatest common factor (GCF) is often the first step in simplifying an expression. It helps to reduce the complexity of the polynomial and makes subsequent factoring steps easier. Now, let's consider the expression x² - 4x + 4. This is a trinomial, and we need to find two numbers that multiply to 4 and add up to -4. These numbers are -2 and -2. Alternatively, we can recognize that this is a perfect square trinomial, which can be factored as (x - 2)². A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recognizing these patterns can save time and effort in factoring. Combining these results, we can write the completely factored form of (6x² + 9x)(x² - 4x + 4) as 3x(2x + 3)(x - 2)(x - 2), which can also be written as 3x(2x + 3)(x - 2)². This example illustrates the importance of combining different factoring techniques to handle more complex expressions. To further enhance your skills, try factoring the following expressions:

• x³ + 8 (Hint: Sum of cubes)
• 2x² - 5x - 3 (Hint: Factoring trinomials)
• x⁴ - 16 (Hint: Difference of squares and difference of squares again)
• 4x² + 12x + 9 (Hint: Perfect square trinomial)

By working through these examples and practice problems, you'll gain a deeper understanding of factoring techniques and develop the ability to apply them effectively. Remember that practice is key to mastering any mathematical skill, and factoring polynomials is no exception. Don't be afraid to make mistakes, as they are valuable learning opportunities. Analyze your errors, understand the underlying concepts, and keep practicing. With consistent effort, you'll become proficient in factoring polynomials and unlock the power of this essential mathematical tool.

Factoring Higher Degree Polynomials

Factoring higher degree polynomials can seem daunting, but with the right strategies, it becomes a manageable task. Let's delve into the techniques for factoring higher degree polynomials, expanding our factoring toolkit. Higher degree polynomials are polynomials with a degree of three or more. Factoring these polynomials often involves a combination of techniques, including factoring by grouping, using the rational root theorem, and synthetic division. One common technique is factoring by grouping. This technique is particularly useful when dealing with polynomials with four or more terms. The idea is to group terms together in such a way that each group has a common factor. For example, consider the polynomial x³ + 2x² - 3x - 6. We can group the first two terms and the last two terms: (x³ + 2x²) + (-3x - 6). Now, we can factor out a common factor from each group: x²(x + 2) - 3(x + 2). Notice that both terms now have a common factor of (x + 2). We can factor out (x + 2) to get (x + 2)(x² - 3). This is the factored form of the polynomial. Another powerful tool for factoring higher degree polynomials is the rational root theorem. The rational root theorem provides a list of possible rational roots of a polynomial. A rational root is a root that can be expressed as a fraction p/q, where p and q are integers. Once we have a list of possible rational roots, we can test them using synthetic division. Synthetic division is a shortcut method for dividing a polynomial by a linear factor. If the remainder is zero, then the linear factor is a factor of the polynomial, and the root is a root of the polynomial. For example, consider the polynomial x³ - 6x² + 11x - 6. The rational root theorem tells us that the possible rational roots are ±1, ±2, ±3, and ±6. We can test these roots using synthetic division. If we test x = 1, we find that it is a root. This means that (x - 1) is a factor of the polynomial. We can use synthetic division to divide x³ - 6x² + 11x - 6 by (x - 1), which gives us the quotient x² - 5x + 6. We can then factor the quadratic x² - 5x + 6 as (x - 2)(x - 3). Therefore, the factored form of the polynomial is (x - 1)(x - 2)(x - 3). Factoring higher degree polynomials requires a combination of skills and techniques. It's essential to be familiar with the various factoring methods and to know when to apply each method. With practice, you'll develop the ability to factor higher degree polynomials efficiently and accurately.

Conclusion: Mastering the Art of Factoring

In conclusion, mastering the art of factoring polynomials is a crucial skill in mathematics. From basic quadratic expressions to higher degree polynomials, the ability to break down expressions into their constituent factors is fundamental to solving equations, simplifying expressions, and analyzing functions. We've explored various techniques, including factoring out the greatest common factor, recognizing the difference of squares, factoring trinomials, and employing methods for higher degree polynomials such as factoring by grouping and the rational root theorem. Factoring is not just a mechanical process; it's a problem-solving skill that requires careful observation, strategic thinking, and a solid understanding of algebraic principles. The benefits of mastering factoring extend far beyond the classroom. In fields such as engineering, physics, and computer science, factoring is used to simplify complex models, solve real-world problems, and develop efficient algorithms. The ability to factor polynomials enhances your mathematical toolkit and empowers you to tackle a wide range of challenges. To continue honing your factoring skills, it's essential to practice regularly and work through a variety of examples. Don't be discouraged by difficult problems; instead, view them as opportunities to learn and grow. Break down complex expressions into smaller parts, identify patterns, and apply the appropriate techniques. Remember, the more you practice, the more intuitive factoring will become. As you delve deeper into mathematics, you'll encounter more advanced concepts that build upon the foundation of factoring. From calculus to linear algebra, the ability to manipulate and simplify polynomial expressions is essential for success. Therefore, investing time and effort in mastering factoring is an investment in your mathematical future. In summary, factoring polynomials is an art that combines mathematical knowledge, strategic thinking, and persistent practice. By mastering this art, you'll unlock the power of algebra and pave the way for advanced mathematical studies and applications. So, embrace the challenge, practice diligently, and enjoy the satisfaction of completely factoring even the most complex expressions.