Simplifying Polynomial Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the world of simplifying polynomial expressions. Polynomials might sound intimidating, but trust me, they're just fancy terms for algebraic expressions with multiple terms. We're going to break down a specific example step-by-step, so you'll be a pro in no time. This guide will walk you through the process of simplifying expressions with exponents and variables, focusing on a clear, easy-to-understand approach. Understanding how to simplify these expressions is crucial for various mathematical and scientific applications, making this a valuable skill to master. So, let’s get started and make polynomials less puzzling and more practical!

Understanding the Problem

Before we jump into solving, let's understand what we're dealing with. Our mission, should we choose to accept it (and you should!), is to simplify this expression:

$$-6 m^5 n^2 \times 2 m^2 n^9 + 15 m^7 n^{11}$$

What does this even mean?

This looks like a jumble of letters and numbers, but it’s actually quite organized. We have two terms here, separated by the addition sign (+). Each term consists of coefficients (the numbers), variables (m and n), and exponents (the little numbers hanging out up high).

• Coefficients: These are the numerical parts of the terms. In our expression, we have -6, 2, and 15 as coefficients. They tell us the magnitude of each term.
• Variables: These are the letters, like m and n, representing unknown values. Variables are the foundation of algebra, allowing us to express relationships and solve for unknowns.
• Exponents: These are the superscripts, like the 5 in $m^5$. They indicate how many times the base (the variable) is multiplied by itself. For example, $m^5$ means m * m * m * m * m.

Why simplify?

Simplifying expressions makes them easier to work with. It’s like tidying up your room – things are just clearer and more manageable. In math, a simplified expression is easier to evaluate, solve, and use in further calculations. A simplified form reduces the chances of making mistakes and makes the underlying structure more apparent. This is particularly important in advanced mathematical contexts and real-world applications.

Our Strategy

To simplify this expression, we'll use the order of operations (PEMDAS/BODMAS) and the rules of exponents. We’ll first handle the multiplication, and then see if we can combine any like terms.

Now that we have a handle on the basics, let’s roll up our sleeves and get to the actual simplification!

Step 1: Multiplying the First Term

The first part of our journey is to tackle the multiplication. We have $-6 m^5 n^2$ multiplied by $2 m^2 n^9$. Let’s break it down into smaller, digestible pieces.

Multiplying Coefficients:

First, we multiply the coefficients: -6 and 2.

$$-6 \times 2 = -12$$

So far, so good! We've handled the numerical part. Now let's move on to the variables.

Multiplying Variables with Exponents:

This is where the exponent rules come into play. When multiplying variables with the same base, we add their exponents. Remember this golden rule: $x^a \times x^b = x^{a+b}$.

• For m: We have $m^5$ and $m^2$. Adding the exponents, we get $m^{5+2} = m^7$.
• For n: We have $n^2$ and $n^9$. Adding those exponents, we get $n^{2+9} = n^{11}$.

Putting It All Together:

Now we combine the results: the coefficient (-12) and the variables with their new exponents ($m^7$ and $n^{11}$). This gives us:

$$-12 m^7 n^{11}$$

Why does this work?

Think of exponents as shorthand for repeated multiplication. $m^5$ is m multiplied by itself five times, and $m^2$ is m multiplied by itself twice. When we multiply $m^5$ and $m^2$, we're essentially multiplying m by itself seven times (5 + 2). This principle holds true for any variable with exponents.

We’ve successfully multiplied the first term! We've navigated the coefficients and the exponents, and now we have a much cleaner term to work with. Next up, we'll incorporate the second term and see if we can simplify further. Keep that momentum going!

Step 2: Rewriting the Expression

Now that we've conquered the first multiplication, let's rewrite our expression with the simplified term. Remember, we started with:

$$-6 m^5 n^2 \times 2 m^2 n^9 + 15 m^7 n^{11}$$

And we simplified the first part to:

$$-12 m^7 n^{11}$$

So, our updated expression looks like this:

$$-12 m^7 n^{11} + 15 m^7 n^{11}$$

What have we done?

We’ve essentially taken a complex part of the expression and made it simpler. This is a crucial step in simplifying any mathematical expression. By breaking it down piece by piece, we avoid getting overwhelmed and reduce the chance of errors. It's like organizing your desk before starting a big project – everything is now more accessible and manageable.

Why is this important?

Rewriting the expression in a simplified form sets us up for the next step: combining like terms. It’s like preparing the ingredients before you start cooking – you want everything ready and in its place. A clear, simplified expression makes it easier to spot patterns and like terms, which is essential for further simplification.

Looking Ahead:

Notice anything interesting about our new expression? We now have two terms that look quite similar. This is a hint that we might be able to combine them. In the next step, we'll focus on identifying and combining these like terms, bringing us even closer to our final simplified expression. So, let's keep moving forward and see how much further we can simplify!

Step 3: Combining Like Terms

Okay, guys, we're getting to the good part! We’ve simplified the multiplication and rewritten the expression. Now, let's see if we can combine some terms. Our expression currently looks like this:

$$-12 m^7 n^{11} + 15 m^7 n^{11}$$

What are 'like terms'?

Like terms are terms that have the same variables raised to the same powers. They’re like the apples and oranges of algebra – you can only combine apples with apples and oranges with oranges. In our case, we need to look for terms that have the same variables (m and n) with the same exponents (7 and 11).

Identifying Like Terms:

Looking at our expression, we have two terms:

• $-12 m^7 n^{11}$
• $15 m^7 n^{11}$

Notice that both terms have $m^7$ and $n^{11}$. This means they are indeed like terms! We can combine them, just like adding 2 apples to 3 apples.

Combining Like Terms:

To combine like terms, we simply add (or subtract) their coefficients. The variables and exponents stay the same. It’s like saying 2 apples + 3 apples = 5 apples – the