Factoring Polynomials What Polynomial Is Represented By (2x+3)(3x+5)

In the fascinating world of algebra, factoring polynomials is a fundamental skill. It's the process of breaking down a polynomial expression into its constituent factors, much like dissecting a composite number into its prime factors. This ability to deconstruct polynomials is not merely an academic exercise; it's a powerful tool that unlocks solutions to a wide array of mathematical problems. From solving quadratic equations to simplifying complex algebraic expressions, polynomial factorization forms the bedrock of advanced mathematical concepts. When we are presented with the factored form of a polynomial, such as $(2x + 3)(3x + 5)$, the challenge lies in reversing the process – in expanding the factored form to reveal the original polynomial expression. This expansion unveils the polynomial that the given factors represent, providing valuable insights into the polynomial's structure and behavior. Understanding how to perform this expansion is crucial for various mathematical applications, including graphing polynomials, finding their roots, and performing algebraic manipulations. In essence, the ability to seamlessly transition between the factored form and the expanded form of a polynomial is a cornerstone of algebraic proficiency.

Understanding the Problem: Decoding the Factored Form

Our specific task is to identify the polynomial that corresponds to the factored form $(2x + 3)(3x + 5)$. This means we need to reverse the factorization process and expand the given expression. The factored form provides a compact representation of the polynomial, highlighting its roots and structure. Each factor, $(2x + 3)$ and $(3x + 5)$, represents a linear expression. When multiplied together, these factors produce a polynomial of higher degree. The degree of the resulting polynomial is determined by the highest power of the variable 'x'. In this case, since we are multiplying two linear expressions (each with 'x' raised to the power of 1), the resulting polynomial will be a quadratic expression (where 'x' is raised to the power of 2). To uncover the polynomial hidden within the factored form, we employ the distributive property, a fundamental principle of algebra that governs how expressions within parentheses are multiplied. This involves systematically multiplying each term in the first factor by each term in the second factor. By carefully applying the distributive property and combining like terms, we can unveil the original polynomial.

Step-by-Step Expansion: Unveiling the Polynomial

The key to expanding the factored form $(2x + 3)(3x + 5)$ lies in the meticulous application of the distributive property. This property dictates that each term in the first factor must be multiplied by each term in the second factor. This can be visualized as a systematic process: First, multiply the first term of the first factor, $2x$, by both terms of the second factor, $3x$ and $5$. This yields $2x * 3x = 6x^2$ and $2x * 5 = 10x$. Next, multiply the second term of the first factor, $3$, by both terms of the second factor, $3x$ and $5$. This gives us $3 * 3x = 9x$ and $3 * 5 = 15$. Now we have four terms: $6x^2$, $10x$, $9x$, and $15$. These terms represent the individual products resulting from the distribution. The next crucial step is to combine like terms. Like terms are those that have the same variable raised to the same power. In our expanded expression, $10x$ and $9x$ are like terms, as they both contain 'x' raised to the power of 1. Combining these terms involves adding their coefficients: $10x + 9x = 19x$. The other terms, $6x^2$ and $15$, do not have any like terms and remain as they are. Finally, we arrange the terms in descending order of their exponents, a convention that ensures clarity and consistency in representing polynomials. This process of expansion and simplification allows us to transform the factored form into its equivalent polynomial representation, revealing the underlying structure of the expression.

The Resulting Polynomial: 6x^2 + 19x + 15

After carefully applying the distributive property and combining like terms, we arrive at the polynomial $6x^2 + 19x + 15$. This is the polynomial that is represented by the factored form $(2x + 3)(3x + 5)$. This resulting polynomial is a quadratic expression, characterized by the highest power of 'x' being 2. The coefficient of the $x^2$ term is 6, the coefficient of the 'x' term is 19, and the constant term is 15. This polynomial encapsulates the same mathematical information as its factored form but presents it in a different way. The expanded form allows us to readily identify the coefficients and the degree of the polynomial, which are crucial for various algebraic manipulations and analyses. For instance, we can easily determine the y-intercept of the polynomial's graph by setting $x = 0$, which yields the constant term, 15. The coefficients also play a vital role in determining the shape and behavior of the polynomial's graph. The factored form, on the other hand, provides direct insight into the roots of the polynomial, which are the values of 'x' that make the polynomial equal to zero. In this case, the roots can be found by setting each factor to zero and solving for 'x'. Understanding the relationship between the factored form and the expanded form of a polynomial is essential for a comprehensive understanding of polynomial behavior and their applications in various mathematical contexts.

Verification and Alternative Methods: Ensuring Accuracy

To ensure the accuracy of our expansion, it's always prudent to employ verification methods. One such method is to substitute a numerical value for 'x' in both the factored form and the expanded form and compare the results. If the two expressions yield the same value for the chosen 'x', it strengthens our confidence in the correctness of the expansion. For instance, let's substitute $x = 1$ into both forms. In the factored form, $(2x + 3)(3x + 5)$ becomes $(2(1) + 3)(3(1) + 5) = (5)(8) = 40$. In the expanded form, $6x^2 + 19x + 15$ becomes $6(1)^2 + 19(1) + 15 = 6 + 19 + 15 = 40$. The fact that both forms evaluate to 40 for $x = 1$ provides strong evidence that our expansion is correct. Another valuable verification technique is to use the FOIL method. FOIL stands for First, Outer, Inner, Last, and it's a mnemonic for remembering the steps of the distributive property when multiplying two binomials (expressions with two terms). The FOIL method provides a structured approach to ensure that all terms are multiplied correctly. "First" refers to multiplying the first terms of each binomial $(2x * 3x = 6x^2)$, "Outer" refers to multiplying the outer terms $(2x * 5 = 10x)$, "Inner" refers to multiplying the inner terms $(3 * 3x = 9x)$, and "Last" refers to multiplying the last terms $(3 * 5 = 15)$. Combining these terms, we get $6x^2 + 10x + 9x + 15$, which simplifies to $6x^2 + 19x + 15$, confirming our earlier result. These verification methods not only enhance our confidence in the answer but also reinforce our understanding of the underlying algebraic principles.

Real-World Applications: Polynomials in Action

The study of polynomials transcends the confines of the classroom and finds practical applications in numerous real-world scenarios. Polynomials are instrumental in modeling various physical phenomena, from the trajectory of a projectile to the growth of a population. Engineers, scientists, and economists rely on polynomial functions to represent and analyze complex systems. In physics, for example, quadratic equations (which are polynomials of degree 2) are used to describe the motion of objects under the influence of gravity. The height of a ball thrown into the air can be accurately modeled using a quadratic polynomial, allowing us to predict its trajectory and landing point. In economics, polynomial functions are used to model cost, revenue, and profit curves, enabling businesses to make informed decisions about pricing and production levels. The relationship between the quantity of goods sold and the total revenue generated can often be represented by a polynomial. Computer graphics, a field that has revolutionized visual media, heavily relies on polynomials to create smooth curves and surfaces. Bezier curves, which are widely used in computer-aided design (CAD) and animation software, are defined using polynomial equations. These curves allow designers and animators to create intricate shapes and movements with precision. Polynomials also play a crucial role in data analysis and statistics. Regression analysis, a technique used to find the best-fitting curve for a set of data points, often involves fitting polynomial functions to the data. This allows us to identify trends and make predictions based on past observations. The applications of polynomials are vast and diverse, highlighting their importance as a fundamental tool in various fields of study and practice.

Conclusion: Mastering Polynomial Factorization

In conclusion, understanding how to factor polynomials and expand factored expressions is a cornerstone of algebraic proficiency. The ability to seamlessly transition between the factored form and the expanded form of a polynomial empowers us to solve a wide range of mathematical problems and gain deeper insights into polynomial behavior. In this particular case, we successfully expanded the factored form $(2x + 3)(3x + 5)$ to reveal the polynomial $6x^2 + 19x + 15$. This process involved the meticulous application of the distributive property, the careful combination of like terms, and the use of verification methods to ensure accuracy. Mastering polynomial factorization is not merely an academic exercise; it's a skill that opens doors to advanced mathematical concepts and real-world applications. From solving equations to modeling physical phenomena, polynomials are ubiquitous in various fields of study and practice. By developing a strong foundation in polynomial algebra, we equip ourselves with the tools necessary to tackle complex problems and make informed decisions in a variety of contexts. The journey of learning algebra is a journey of discovery, and the mastery of polynomial factorization is a significant milestone along that path.