Determining Binomial Results From The Operation (3y^6+4)(9y^12-12y^6+16)

Introduction

In the realm of mathematics, particularly in algebra, understanding different operations and their results is crucial. One such concept is the binomial, a polynomial expression with exactly two terms. Identifying which operations result in a binomial is a fundamental skill. This article delves into a specific operation, $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$, to determine whether it results in a binomial. We will explore the underlying principles, step-by-step calculations, and the broader context of polynomial operations. This comprehensive guide aims to provide clarity and insight into the process, ensuring a solid understanding of how to approach similar problems. By breaking down the expression and analyzing its components, we can confidently ascertain the final result and its classification within algebraic expressions. The process involves applying the distributive property and simplifying the terms to arrive at a conclusive answer. This exploration not only answers the specific question but also enhances your overall understanding of algebraic manipulations and polynomial forms. We'll start by defining what a binomial is and then move on to dissecting the given expression. Understanding the terminology and the methods will be key to mastering these types of problems. So, let's embark on this mathematical journey to unravel the complexities of polynomial operations and binomial outcomes.

What is a Binomial?

Before we dive into the operation $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$, it’s essential to define what a binomial actually is. In simple terms, a binomial is a polynomial expression that consists of exactly two terms. These terms are typically connected by an addition or subtraction operation. For example, $x + y$, $2a - 3b$, and $5m^2 + 7n$ are all binomials. Each of these expressions contains two distinct terms. The terms can involve variables, constants, or a combination of both, but the key is that there are only two terms present in the expression. Understanding this definition is crucial because it sets the foundation for identifying whether a given operation results in a binomial. When we talk about terms, we refer to parts of an expression that are separated by addition or subtraction. For instance, in the binomial $2x + 3$, $2x$ and $3$ are the two terms. Similarly, in the binomial $4y^2 - 5y$, $4y^2$ and $-5y$ are the two terms. This distinction is important because it helps us differentiate binomials from other types of polynomials, such as monomials (one term), trinomials (three terms), and polynomials with more than three terms. Identifying binomials correctly is a fundamental skill in algebra and is often used in various mathematical contexts, including factoring, expanding expressions, and solving equations. Now that we have a clear understanding of what a binomial is, we can proceed to analyze the given operation and determine whether it fits this definition.

Dissecting the Operation: $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$

The given operation is $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$. This expression involves the product of a binomial and a trinomial (a polynomial with three terms). To determine whether this operation results in a binomial, we need to perform the multiplication and simplify the resulting expression. This process involves applying the distributive property, which states that each term in the first polynomial must be multiplied by each term in the second polynomial. In this case, we will multiply each term of the binomial $\left(3 y^6+4\right)$ by each term of the trinomial $\left(9 y^{12}-12 y^6+16\right)$. This will result in a series of terms that we will then combine and simplify. Breaking down the expression in this way allows us to manage the complexity and systematically arrive at the correct result. The distributive property is a cornerstone of algebraic manipulation, and its proper application is essential for accurately expanding and simplifying polynomial expressions. By meticulously multiplying each term and keeping track of the exponents and coefficients, we can avoid errors and ensure the final expression is in its simplest form. This step-by-step approach is not only crucial for this specific problem but also for handling more complex polynomial operations in the future. We will now proceed with the multiplication, carefully applying the distributive property to each term.

Step-by-Step Calculation

To perform the multiplication $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$, we apply the distributive property. First, we multiply $3y^6$ by each term in the trinomial: $3y^6 * 9y^{12} = 27y^{18}$, $3y^6 * -12y^6 = -36y^{12}$, and $3y^6 * 16 = 48y^6$. Next, we multiply $4$ by each term in the trinomial: $4 * 9y^{12} = 36y^{12}$, $4 * -12y^6 = -48y^6$, and $4 * 16 = 64$. Now, we combine these terms: $27y^{18} - 36y^{12} + 48y^6 + 36y^{12} - 48y^6 + 64$. Notice that some terms are like terms, meaning they have the same variable and exponent. Specifically, $-36y^{12}$ and $36y^{12}$ are like terms, and $48y^6$ and $-48y^6$ are like terms. These like terms can be combined to simplify the expression. The process of combining like terms involves adding or subtracting their coefficients while keeping the variable and exponent the same. This simplification is a critical step in determining the final form of the expression and whether it meets the criteria of a binomial. By meticulously performing each multiplication and then combining like terms, we ensure an accurate result. This methodical approach is crucial in algebra to avoid errors and to simplify complex expressions effectively. In the next step, we will combine the like terms to see what our simplified expression looks like.

Simplifying the Expression

After performing the initial multiplication, we have the expression: $27y^{18} - 36y^{12} + 48y^6 + 36y^{12} - 48y^6 + 64$. Now, we simplify by combining like terms. We have $-36y^{12}$ and $+36y^{12}$, which cancel each other out $(-36y^{12} + 36y^{12} = 0)$. Similarly, we have $+48y^6$ and $-48y^6$, which also cancel each other out $(48y^6 - 48y^6 = 0)$. This leaves us with the simplified expression: $27y^{18} + 64$. This simplified expression consists of two terms: $27y^{18}$ and $64$. Since there are exactly two terms, this expression is indeed a binomial. The process of simplification is crucial in algebra, as it allows us to reduce complex expressions to their most basic form, making them easier to understand and work with. In this case, the cancellation of like terms significantly simplified the expression, revealing its true nature as a binomial. This result highlights the importance of careful calculation and simplification when dealing with polynomial operations. By identifying and combining like terms, we can often unveil the underlying structure of an expression and determine its classification within algebraic terms. The simplified expression $27y^{18} + 64$ clearly fits the definition of a binomial, as it consists of two distinct terms connected by an addition operation. Now that we have simplified the expression, we can confidently conclude whether the original operation results in a binomial.

Conclusion: Does the Operation Result in a Binomial?

After performing the multiplication $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$ and simplifying the result, we arrived at the expression $27y^{18} + 64$. As we have established, a binomial is a polynomial expression with exactly two terms. The simplified expression $27y^{18} + 64$ has two terms: $27y^{18}$ and $64$. Therefore, the operation $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$ does indeed result in a binomial. This conclusion is reached through a methodical process of applying the distributive property, multiplying terms, and simplifying the resulting expression by combining like terms. Understanding these steps is crucial for tackling similar problems in algebra. The ability to recognize and manipulate polynomial expressions is a fundamental skill in mathematics, and this example provides a clear illustration of how to determine the result of a specific operation. The process not only confirms whether an expression is a binomial but also reinforces the importance of careful calculation and simplification in algebraic manipulations. This knowledge is applicable in various mathematical contexts, from solving equations to simplifying complex expressions. Therefore, mastering this process is an essential step in building a strong foundation in algebra.

In summary, the operation $\left(3 y^6+4\right)\left(9 y^{12}-12 y^6+16\right)$ results in a binomial: $27y^{18} + 64$.