Factoring Polynomials How To Find The Factored Form Of X² - 16x + 48

In the realm of algebra, factoring polynomials is a fundamental skill. It involves breaking down a polynomial expression into simpler expressions that, when multiplied together, give the original polynomial. This process is not only crucial for solving equations but also for simplifying complex expressions and understanding the behavior of functions. In this comprehensive guide, we will delve into the process of factoring the quadratic polynomial x² - 16x + 48. We will explore the underlying principles, step-by-step methods, and various techniques to arrive at the factored form of the given polynomial. Whether you are a student grappling with algebra concepts or someone looking to refresh your mathematical skills, this guide will provide you with a clear and concise understanding of polynomial factorization.

Understanding Polynomials and Factoring

Before we dive into the specific example, let's establish a solid foundation by understanding what polynomials are and why factoring is so important. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples of polynomials include x² + 3x + 2, 2x³ - 5x² + x - 7, and the one we'll be working with today, x² - 16x + 48. Factoring, in essence, is the reverse process of expansion. When we expand, we multiply expressions together to get a larger expression. Factoring, on the other hand, involves finding the expressions that multiply together to give us the polynomial we started with. Think of it like this: if expansion is like building a house from bricks, factoring is like taking the house apart to see the individual bricks that make it up. Factoring is crucial for several reasons. First and foremost, it helps us solve polynomial equations. When a polynomial is set equal to zero, factoring allows us to find the values of the variable that make the equation true, also known as the roots or zeros of the polynomial. Furthermore, factoring simplifies complex expressions, making them easier to work with in various mathematical operations. It also provides insights into the behavior of the polynomial function, such as its intercepts and turning points. In the next sections, we will explore the techniques used to factor quadratic polynomials like x² - 16x + 48 and understand the logic behind each step. By mastering these techniques, you'll be well-equipped to tackle a wide range of polynomial factorization problems.

Identifying the Type of Polynomial

The first step in factoring a polynomial is to identify the type of polynomial you are dealing with. This will help you determine the most appropriate factoring method to use. Our polynomial, x² - 16x + 48, is a quadratic polynomial. A quadratic polynomial is a polynomial of degree 2, meaning the highest power of the variable (in this case, x) is 2. The general form of a quadratic polynomial is ax² + bx + c, where a, b, and c are constants. In our case, a = 1, b = -16, and c = 48. Recognizing that our polynomial is quadratic is crucial because it allows us to employ specific factoring techniques tailored for quadratic expressions. There are several methods for factoring quadratics, including: Factoring by grouping: This method involves rewriting the middle term (bx) as the sum of two terms and then grouping terms to factor out common factors. The quadratic formula: This formula provides the solutions (roots) of a quadratic equation and can be used to derive the factored form. Completing the square: This technique involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored easily. The method we will focus on in this guide is factoring by finding two numbers that add up to the coefficient of the x term (b) and multiply to the constant term (c). This method is particularly effective for quadratics where the leading coefficient (a) is 1, as in our case. In the following sections, we will delve into this method and apply it to factor x² - 16x + 48. Understanding the structure of quadratic polynomials and the available factoring techniques is the key to successfully factoring them. So, let's move on to the next step: finding the right numbers that will lead us to the factored form of our polynomial.

Finding the Right Numbers: The Key to Factoring

Now that we've identified our polynomial as a quadratic of the form x² - 16x + 48, with a = 1, b = -16, and c = 48, we can employ the method of finding two numbers that add up to b and multiply to c. This is a crucial step in factoring quadratics where the leading coefficient is 1. The logic behind this method stems from the expansion of factored quadratic expressions. When we expand an expression like (x + p)(x + q), we get x² + (p + q)x + pq. Notice that the coefficient of the x term is the sum of p and q, and the constant term is the product of p and q. Therefore, to factor a quadratic like x² - 16x + 48, we need to find two numbers, let's call them p and q, that satisfy the following conditions: p + q = -16 (the coefficient of the x term) p * q = 48 (the constant term) To find these numbers, we can start by listing the factors of 48. Since the product is positive and the sum is negative, we know that both numbers must be negative. Here are the pairs of negative factors of 48: -1 and -48 -2 and -24 -3 and -16 -4 and -12 -6 and -8 Now, we need to check which of these pairs adds up to -16. Let's examine each pair: -1 + (-48) = -49 -2 + (-24) = -26 -3 + (-16) = -19 -4 + (-12) = -16 -6 + (-8) = -14 We can see that the pair -4 and -12 adds up to -16, which is the coefficient of our x term. Therefore, -4 and -12 are the numbers we've been looking for. These numbers are the key to unlocking the factored form of our polynomial. In the next section, we will use these numbers to construct the factored expression and verify our result. By systematically searching for the right numbers, we can effectively factor quadratic polynomials and simplify complex algebraic expressions.

Constructing the Factored Form

With the numbers -4 and -12 identified as the pair that adds up to -16 and multiplies to 48, we are now ready to construct the factored form of the polynomial x² - 16x + 48. Recall that our goal is to rewrite the polynomial as a product of two binomials. Since the leading coefficient of our quadratic is 1, the factored form will have the general structure of (x + p)(x + q), where p and q are the numbers we found. In our case, p = -4 and q = -12. Therefore, we can directly substitute these values into the binomial factors: (x + (-4))(x + (-12)) Simplifying the expressions inside the parentheses, we get: (x - 4)(x - 12) This is the factored form of the polynomial x² - 16x + 48. To ensure we have factored correctly, it's always a good practice to expand the factored form and check if it matches the original polynomial. Let's expand (x - 4)(x - 12) using the distributive property (also known as FOIL): (x - 4)(x - 12) = x(x - 12) - 4(x - 12) = x² - 12x - 4x + 48 = x² - 16x + 48 As we can see, the expanded form matches our original polynomial, x² - 16x + 48. This confirms that our factoring is correct. The factored form (x - 4)(x - 12) represents the polynomial as a product of two linear factors. This form is not only useful for solving quadratic equations but also provides insights into the roots and behavior of the quadratic function. In the next section, we will discuss the significance of the factored form and its applications in solving equations and understanding the graphical representation of the polynomial. By mastering the process of constructing the factored form, you gain a powerful tool for analyzing and manipulating polynomial expressions.

Significance of the Factored Form

The factored form of a polynomial, such as (x - 4)(x - 12) for the polynomial x² - 16x + 48, is not just a different way of writing the same expression; it carries significant information and has practical applications in various mathematical contexts. One of the most important uses of the factored form is in solving quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0. To solve such an equation, we can first factor the quadratic expression (if possible) and then use the zero-product property. The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. For example, to solve the equation x² - 16x + 48 = 0, we can use the factored form (x - 4)(x - 12) = 0. Applying the zero-product property, we set each factor equal to zero: x - 4 = 0 or x - 12 = 0 Solving these equations, we find x = 4 and x = 12. These are the solutions (or roots) of the quadratic equation. The roots represent the x-intercepts of the graph of the quadratic function y = x² - 16x + 48. Graphically, the factored form reveals where the parabola intersects the x-axis. The roots, 4 and 12, are the x-coordinates of these intersection points. Furthermore, the factored form can help us understand the behavior of the quadratic function. For instance, the signs of the factors (x - 4) and (x - 12) tell us where the function is positive or negative. This information can be used to sketch the graph of the parabola and determine its key features, such as the vertex and axis of symmetry. In addition to solving equations and graphing functions, the factored form is also useful in simplifying rational expressions and performing other algebraic manipulations. By understanding the significance of the factored form, we can leverage it as a powerful tool in a wide range of mathematical problems. It provides valuable insights into the properties and behavior of polynomials, making it an essential concept in algebra.

Conclusion

In this comprehensive guide, we have explored the process of factoring the polynomial x² - 16x + 48. We began by understanding the importance of factoring polynomials in algebra and its role in solving equations and simplifying expressions. We then identified our polynomial as a quadratic polynomial and discussed the general methods for factoring quadratics. The key to factoring x² - 16x + 48 lay in finding two numbers that add up to -16 and multiply to 48. Through a systematic search, we identified -4 and -12 as the correct numbers. These numbers allowed us to construct the factored form of the polynomial as (x - 4)(x - 12). We verified our result by expanding the factored form and confirming that it matched the original polynomial. Finally, we discussed the significance of the factored form, highlighting its use in solving quadratic equations and understanding the graphical representation of the polynomial. The factored form not only provides the roots of the equation but also offers insights into the behavior of the quadratic function. Mastering the technique of factoring polynomials is a fundamental skill in algebra. It equips you with a powerful tool for solving equations, simplifying expressions, and analyzing functions. By understanding the underlying principles and practicing various factoring techniques, you can confidently tackle a wide range of algebraic problems. Remember, the key to successful factoring is a systematic approach, a clear understanding of the concepts, and consistent practice. As you continue your journey in mathematics, the skills you've gained in this guide will serve you well in more advanced topics and applications.