Simplifying Polynomial Expression X²Y - Z

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In this article, we will delve into the fascinating world of polynomial expressions and explore how to simplify them. We are given three variables, $X$, $Y$, and $Z$, each representing a polynomial in terms of the variable $a$. Our goal is to find the simplest form of the expression $X^2 Y - Z$. This involves substituting the given polynomials for $X$, $Y$, and $Z$, expanding the resulting expression, and combining like terms. This exercise is a fundamental concept in algebra, building a solid foundation for more advanced topics. Mastering these simplification techniques is crucial for success in various mathematical fields, including calculus, linear algebra, and differential equations. Furthermore, the ability to manipulate and simplify polynomial expressions is highly valuable in diverse real-world applications, ranging from engineering and physics to economics and computer science. Let's embark on this mathematical journey and discover the beauty of polynomial simplification!

Defining the Variables

Before we dive into the simplification process, let's clearly define the polynomials represented by the variables $X$, $Y$, and $Z$. These variables serve as placeholders for specific algebraic expressions, allowing us to manipulate them more easily. We are given the following definitions:

  • X=aX = a

  • Y=3a−5Y = 3a - 5

  • Z=a2+2Z = a^2 + 2

Here, $X$ is simply equal to the variable $a$, while $Y$ is a linear polynomial involving $a$, and $Z$ is a quadratic polynomial in $a$. Understanding these definitions is paramount, as they form the basis for our subsequent calculations. We will be substituting these expressions into the given expression $X^2 Y - Z$ and then simplifying the result. The process of substitution is a cornerstone of algebraic manipulation, enabling us to rewrite expressions in different forms and reveal hidden relationships between variables. By carefully substituting and expanding, we can transform complex expressions into simpler, more manageable forms. This skill is essential for solving equations, analyzing functions, and tackling a wide range of mathematical problems. So, let's keep these definitions firmly in mind as we proceed with our simplification endeavor.

Substituting the Polynomials

Now that we have clearly defined the variables $X$, $Y$, and $Z$, the next step is to substitute their respective polynomial expressions into the given expression $X^2 Y - Z$. This substitution process is crucial for transforming the expression into a form that we can simplify. Replacing each variable with its corresponding polynomial, we get:

X2Y−Z=(a)2(3a−5)−(a2+2)X^2 Y - Z = (a)^2 (3a - 5) - (a^2 + 2)

Notice how we have replaced $X$ with $a$, $Y$ with $(3a - 5)$, and $Z$ with $(a^2 + 2)$. It's important to use parentheses carefully to maintain the correct order of operations and ensure that the substitution is performed accurately. Failing to do so can lead to errors in the subsequent steps. This substitution step is a fundamental technique in algebra, allowing us to rewrite expressions in terms of a single variable or a set of variables. By performing this substitution, we have effectively transformed the original expression into a polynomial expression involving only the variable $a$. This sets the stage for the next step, where we will expand the expression and combine like terms to obtain the simplest form. So, let's proceed with the expansion and unravel the expression further.

Expanding the Expression

With the polynomials substituted, our expression now looks like this:

(a)2(3a−5)−(a2+2)(a)^2 (3a - 5) - (a^2 + 2)

The next step is to expand this expression by performing the necessary multiplications. First, we need to simplify $(a)^2$, which is simply $a^2$. Then, we multiply $a^2$ by the binomial $(3a - 5)$, and we distribute the negative sign across the terms inside the second set of parentheses. This gives us:

a2(3a−5)−(a2+2)=a2(3a)+a2(−5)−a2−2a^2 (3a - 5) - (a^2 + 2) = a^2(3a) + a^2(-5) - a^2 - 2

=3a3−5a2−a2−2= 3a^3 - 5a^2 - a^2 - 2

Here, we have used the distributive property to multiply $a^2$ by each term inside the parentheses. The distributive property is a fundamental concept in algebra, allowing us to multiply a single term by a sum or difference of terms. This step is crucial for removing the parentheses and transforming the expression into a sum of individual terms. It's essential to pay close attention to the signs when expanding expressions, as a simple sign error can lead to an incorrect final result. By carefully applying the distributive property, we have successfully expanded the expression, preparing it for the final step of combining like terms. This process of expansion is a cornerstone of algebraic manipulation, enabling us to simplify complex expressions and solve equations effectively. Let's move on to the final step and combine the like terms to achieve the simplest form.

Combining Like Terms

After expanding the expression, we have:

3a3−5a2−a2−23a^3 - 5a^2 - a^2 - 2

Now, we need to identify and combine the like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two terms involving $a^2$: $-5a^2$ and $-a^2$. Combining these terms, we get:

−5a2−a2=−6a2-5a^2 - a^2 = -6a^2

Substituting this back into the expression, we obtain the simplified form:

3a3−6a2−23a^3 - 6a^2 - 2

This is the simplest form of the expression $X^2 Y - Z$. We have successfully combined the like terms, resulting in a concise and easy-to-understand polynomial expression. The process of combining like terms is a crucial step in simplifying algebraic expressions. It allows us to reduce the number of terms and obtain a more compact representation of the expression. This simplification not only makes the expression easier to work with but also reveals its underlying structure and properties. The ability to identify and combine like terms is a fundamental skill in algebra, essential for solving equations, simplifying formulas, and tackling a wide range of mathematical problems. By carefully combining the like terms, we have arrived at the final simplified form of the expression, demonstrating the power of algebraic manipulation.

Final Answer

Therefore, the expression $X^2 Y - Z$ in its simplest form is:

3a3−6a2−2\boxed{3a^3 - 6a^2 - 2}

We have successfully navigated the process of simplifying a polynomial expression, starting with the substitution of variables, expanding the expression, and finally combining like terms. This journey highlights the fundamental principles of algebra and demonstrates how these principles can be applied to solve mathematical problems. The ability to simplify expressions is a crucial skill in mathematics, paving the way for more advanced topics and applications. From solving equations to analyzing functions, the techniques we have employed here are widely applicable across various mathematical domains. Furthermore, these skills extend beyond the realm of pure mathematics, finding practical use in fields such as engineering, physics, and computer science. So, let's carry this knowledge forward and continue to explore the fascinating world of mathematics. Congratulations on mastering this simplification process!