Simplifying Polynomial Expressions A Step-by-Step Guide

Simplifying algebraic expressions is a fundamental skill in mathematics, especially when dealing with polynomials. Polynomials are expressions containing variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In this article, we will delve into the process of simplifying a specific polynomial expression. We aim to understand the underlying principles and techniques involved. Our focus will be on the expression $10m(m^2 + m + 10)$, which we aim to expand and simplify. This involves distributing the term outside the parenthesis, $10m$, across each term inside the parenthesis. This process is crucial in various mathematical contexts, such as solving equations, graphing functions, and performing calculus operations. By mastering the simplification of polynomial expressions, you build a solid foundation for more advanced mathematical concepts. This guide will not only provide a step-by-step solution but also explain the reasoning behind each step, ensuring a thorough understanding of the topic.

Understanding the Distributive Property

The distributive property is a cornerstone of algebraic manipulation, and it's essential for simplifying expressions like the one we're tackling. This property states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. In simpler terms, it means that multiplying a number by a sum is the same as multiplying the number by each term in the sum individually and then adding the products. This property extends to expressions with more than two terms inside the parenthesis, such as our expression $10m(m^2 + m + 10)$.

When we apply the distributive property to this expression, we multiply $10m$ by each term inside the parenthesis: $m^2$, $m$, and $10$. This gives us $10m * m^2 + 10m * m + 10m * 10$. Each of these multiplications involves multiplying coefficients (the numerical parts) and variables separately. For the variables, we use the rule of exponents that states $x^a * x^b = x^{a+b}$. Understanding and correctly applying the distributive property is crucial for expanding and simplifying polynomial expressions. It allows us to break down complex expressions into simpler terms that can be easily combined.

Step-by-Step Expansion of the Expression

Let's walk through the step-by-step expansion of the expression $10m(m^2 + m + 10)$. The first step is to distribute the term $10m$ across each term inside the parenthesis. This means we multiply $10m$ by $m^2$, $m$, and $10$ individually.

1. Multiply $10m$ by $m^2$: This gives us $10m * m^2$. Using the exponent rule $x^a * x^b = x^{a+b}$, we get $10m^{1+2} = 10m^3$.
2. Multiply $10m$ by $m$: This gives us $10m * m$. Again, applying the exponent rule, we get $10m^{1+1} = 10m^2$.
3. Multiply $10m$ by $10$: This gives us $10m * 10$. Multiplying the coefficients, we get $100m$.

Now, we combine these results to form the expanded expression: $10m^3 + 10m^2 + 100m$. This expression represents the original expression in its expanded form. It's crucial to perform each multiplication carefully and correctly to avoid errors in the final result. This step-by-step approach makes the process clear and manageable, even for those new to polynomial manipulation.

Identifying Like Terms

After expanding the expression, the next crucial step in simplification is to identify like terms. Like terms are terms that have the same variable raised to the same power. In the expanded expression $10m^3 + 10m^2 + 100m$, each term has a different power of $m$: $m^3$, $m^2$, and $m$ (which is $m^1$).

Since there are no terms with the same power of $m$, there are no like terms in this expression. This means that the expression is already in its simplest form after expansion. In other expressions, you might find like terms that can be combined by adding or subtracting their coefficients. For example, in the expression $3x^2 + 5x - 2x^2 + x$, $3x^2$ and $-2x^2$ are like terms, as are $5x$ and $x$. Recognizing like terms is essential for further simplification, as it allows you to reduce the number of terms in the expression and make it more manageable.

Combining Like Terms (Not Applicable in This Case)

Combining like terms is a fundamental step in simplifying algebraic expressions, but in our specific case, it's not applicable. This is because, as we identified in the previous section, the expanded expression $10m^3 + 10m^2 + 100m$ does not contain any like terms. Each term has a unique power of the variable $m$: $m^3$, $m^2$, and $m^1$.

However, it's crucial to understand how to combine like terms in other scenarios. When like terms are present, you can combine them by adding or subtracting their coefficients while keeping the variable and its exponent the same. For instance, if we had an expression like $5x^2 + 3x^2$, we would combine the like terms $5x^2$ and $3x^2$ by adding their coefficients: $5 + 3 = 8$. The result would be $8x^2$. Similarly, in the expression $7y - 2y$, we would subtract the coefficients: $7 - 2 = 5$, resulting in $5y$. Although we don't need to combine like terms in our current problem, understanding this process is vital for simplifying a wide range of algebraic expressions. It helps to reduce complex expressions to their simplest forms, making them easier to work with in further mathematical operations.

The Simplified Expression

After expanding the expression $10m(m^2 + m + 10)$ and checking for like terms, we arrive at the simplified expression: $10m^3 + 10m^2 + 100m$. This is the final answer because there are no like terms to combine, and the expression is fully expanded. The original form of the expression, $10m(m^2 + m + 10)$, is in a factored form, while the simplified form is in an expanded form.

Both forms are equivalent, meaning they represent the same value for any given value of $m$. However, the simplified form is often more useful for certain mathematical operations, such as evaluating the expression for specific values of $m$, finding its derivative or integral in calculus, or analyzing its behavior as a function. Understanding how to expand and simplify expressions like this is a fundamental skill in algebra. It allows you to manipulate expressions into forms that are easier to work with and understand. The ability to simplify expressions is a cornerstone of more advanced mathematical concepts and applications.

Completing the Original Problem

Now, let's revisit the original problem presented in the question. We were given the expression $10m(m^2 + m + 10)$ and asked to simplify it, filling in the missing term in the expression 10m^3 + 10m^2 + oxed{ ext{?}}m. We've already gone through the process of expanding and simplifying the expression, and we found that the simplified form is $10m^3 + 10m^2 + 100m$.

By comparing this simplified form with the given incomplete expression, we can easily identify the missing term. The given expression has $10m^3$ and $10m^2$, which match the first two terms of our simplified expression. The missing term is the coefficient of $m$, which is $100$. Therefore, the completed expression is $10m^3 + 10m^2 + 100m$. This exercise highlights the importance of simplifying expressions to match them with other forms or to identify missing components. It's a common task in algebra and other areas of mathematics, and mastering the simplification process makes it much easier to solve such problems. The final answer to complete the original problem is $100$.

Conclusion

In conclusion, simplifying the expression $10m(m^2 + m + 10)$ involves expanding it using the distributive property and then combining like terms, if any. In this specific case, the expansion resulted in $10m^3 + 10m^2 + 100m$, and since there were no like terms to combine, this was the final simplified form. We also successfully identified the missing term in the original problem, demonstrating the practical application of simplification skills.

Understanding how to simplify algebraic expressions is a crucial skill in mathematics. It forms the basis for more advanced topics and is essential for solving equations, graphing functions, and performing calculations in various fields. By mastering these fundamental techniques, you build a strong foundation for future mathematical endeavors. This guide has provided a comprehensive explanation of the simplification process, emphasizing the importance of each step and the underlying principles involved. With practice, you can confidently tackle a wide range of algebraic expressions and simplify them efficiently and accurately.