Simplify This Polynomial Expression

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics with a problem that might look a little intimidating at first glance, but trust me, it's totally doable. We're talking about finding an equivalent polynomial expression for a given one. Think of it like solving a puzzle where you need to rearrange and combine terms to find the simplest, neatest form. So, grab your pencils, get comfy, and let's break down this expression: $13p - 3p(1 - 2m) - m(2m - 5p) - m^2$. Our goal is to simplify this bad boy and see which of the given options it matches. We'll go step-by-step, making sure we don't miss any crucial details. Remember, when dealing with polynomials, the key is to distribute correctly and combine like terms. It's all about patience and precision, and before you know it, you'll have the answer staring you right in the face. Let's get this math party started!

Understanding Polynomial Expressions and Equivalence

Alright, let's get down to business, fam. What exactly are we trying to achieve here? We're given a polynomial expression, which is basically a mathematical phrase involving variables (like $p$ and $m$ here) and coefficients, combined using addition, subtraction, and multiplication. The key thing about polynomials is that they don't involve division by variables or variables raised to negative powers. Our specific expression, $13p - 3p(1 - 2m) - m(2m - 5p) - m^2$, has a few terms that need a bit of tidying up. We've got multiplication happening within parentheses, which means we need to use the distributive property. This property is your best friend when simplifying expressions like this. It states that $a(b + c) = ab + ac$. We'll be applying this to $ - 3p(1 - 2m)$ and $ - m(2m - 5p)$. After we distribute, we'll be left with a longer polynomial, but the terms won't be organized. That's where combining like terms comes in. Like terms are terms that have the exact same variables raised to the exact same powers. For example, $5x^2y$ and $-2x^2y$ are like terms because they both have $x^2$ and $y$. We can add or subtract their coefficients to combine them. Our mission is to simplify the given expression until it matches one of the options: A, B, C, or D. This means the final expression should be equivalent to the original one, just in a much cleaner, simpler form. We're not changing the value of the expression, just how it looks. Think of it as taking a messy room and organizing it – everything is still there, but it's much easier to see what's what. So, let's roll up our sleeves and tackle that expression!

Step-by-Step Simplification

Okay, let's get our hands dirty with the actual simplification process, shall we? Our starting point is: $13p - 3p(1 - 2m) - m(2m - 5p) - m^2$. The first thing we need to do is tackle those terms in parentheses using the distributive property. Let's start with $-3p(1 - 2m)$. When we distribute $-3p$ to both $1$ and $-2m$, we get:

$$-3p \times 1 = -3p$$

and

$$-3p \times -2m = +6pm$$

So, $-3p(1 - 2m)$ becomes $-3p + 6pm$. Easy peasy, right?

Now, let's move on to the next part: $-m(2m - 5p)$. We distribute $-m$ to both $2m$ and $-5p$:

$$-m \times 2m = -2m^2$$

and

$$-m \times -5p = +5mp$$

So, $-m(2m - 5p)$ becomes $-2m^2 + 5mp$. We're making progress, guys!

Now, let's substitute these back into our original expression. Remember, the original expression was $13p - 3p(1 - 2m) - m(2m - 5p) - m^2$.

It now looks like this: $13p + (-3p + 6pm) + (-2m^2 + 5mp) - m^2$.

Let's rewrite that without the extra parentheses: $13p - 3p + 6pm - 2m^2 + 5mp - m^2$.

We're getting closer! The next crucial step is to combine like terms. We need to scan through this expression and group terms that have the same variables with the same powers.

First, let's look at the terms with just $p$: We have $13p$ and $-3p$. Combining these gives us $(13 - 3)p = 10p$.

Next, let's look at the terms with $pm$ (or $mp$, since multiplication is commutative, $pm$ is the same as $mp$): We have $6pm$ and $5mp$. Combining these gives us $(6 + 5)mp = 11mp$.

Finally, let's look at the terms with $m^2$: We have $-2m^2$ and $-m^2$ (which is the same as $-1m^2$). Combining these gives us $(-2 - 1)m^2 = -3m^2$.

Now, let's put all our combined terms together. We have $10p$, $11mp$, and $-3m^2$. So, the simplified expression is $10p + 11mp - 3m^2$.

It's super important to write it in a standard form, usually with the highest power term first. So, we can rewrite this as $-3m^2 + 11mp + 10p$.

This is our final simplified expression. Let's see which option it matches!

Comparing with the Options

Alright, we've done the hard work and simplified the given polynomial expression to $-3m^2 + 11mp + 10p$. Now it's time to compare this result with the options provided to see which one is the equivalent polynomial expression. Remember, finding an equivalent expression means we've simplified it correctly and it should match one of the choices. Let's list our simplified expression and the given options side-by-side:

Our simplified expression: $-3m^2 + 11mp + 10p$

Option A: $-3m^2 - 15mp + 10p$ Option B: $-3m^2 + 11mp + 10p$ Option C: $-3m^2 - 11mp + 10p$ Option D: $-3m^2 + 15mp + 10p$

Now, let's do a direct comparison. We look at each term in our simplified expression and see if it matches the corresponding term in each option.

• The $m^2$ term: Our expression has $-3m^2$. Options A, B, C, and D all have $-3m^2$. So far, so good for all options!
• The $mp$ term: Our expression has $+11mp$. Option A has $-15mp$, Option B has $+11mp$, Option C has $-11mp$, and Option D has $+15mp$. Aha! Only Option B matches our $+11mp$ term.
• The $p$ term: Our expression has $+10p$. Options A, B, C, and D all have $+10p$. This term matches across the board.

Since Option B is the only one that matches all three terms of our simplified expression ($-3m^2$, $+11mp$, and $+10p$), it is the correct equivalent polynomial expression. We nailed it, guys!

Final Thoughts and Key Takeaways

So there you have it, mathletes! We successfully tackled a polynomial expression that looked a bit complex at first glance and simplified it down to its most basic form. The key takeaway here is that with polynomials, distributing and combining like terms are your superpowers. Never be afraid of those parentheses; they just signal that you need to apply the distributive property. Remember that $a(b+c) = ab + ac$, and that combining like terms means adding or subtracting coefficients of terms with the identical variable parts (like $x^2y$ and $-5x^2y$). We saw that our original expression, $13p - 3p(1 - 2m) - m(2m - 5p) - m^2$, when simplified, became $-3m^2 + 11mp + 10p$. This allowed us to confidently select Option B as the correct answer.

It's also super helpful to write your final simplified polynomial in a standard order, usually from the highest degree term to the lowest. In our case, $m^2$ has a degree of 2, $mp$ has a degree of 2 (1 for $m$ + 1 for $p$), and $p$ has a degree of 1. So, $-3m^2 + 11mp + 10p$ is a nicely ordered expression. It's all about practice, guys. The more you work through these problems, the more comfortable you'll become with the steps and the quicker you'll be able to spot the correct equivalent expressions. Keep those math brains sharp, and remember that every complex problem can be broken down into simpler, manageable steps. Thanks for joining us at Plastik Magazine for this math adventure. Keep exploring, keep learning, and we'll catch you in the next one!