Simplifying Polynomials: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into some math today, specifically, how to simplify the polynomial expression: $2(x-3)^2+5(x-3)-3$. Don't worry, it looks more intimidating than it actually is. We're going to break it down step-by-step, making it super easy to understand. By the end of this, you'll be a pro at simplifying polynomials and feel confident with this type of math problem. So, grab your pencils (or pens!), and let's get started. We're going to transform the given expression into a standard form, which is basically writing it in the form of $ax^2 + bx + c$, where a, b, and c are constants. This will make the expression much easier to understand and use in further calculations. This process involves a bit of algebra, but I promise, it's not as scary as it sounds. We'll be using the distributive property, the FOIL method (if you're familiar with it), and combining like terms. Let's make this fun, shall we? You've got this!

Step 1: Expand the Squared Term

First things first, we need to deal with the $(x-3)^2$ term. Remember that squaring something means multiplying it by itself. So, $(x-3)^2$ is the same as $(x-3)(x-3)$. Now, let's expand this using the FOIL method (First, Outer, Inner, Last), or simply by distributing each term. Multiplying the first terms, we get $x * x = x^2$. Then, for the outer terms, we have $x * -3 = -3x$. Next, we multiply the inner terms: $-3 * x = -3x$. Finally, we multiply the last terms: $-3 * -3 = 9$. Now we have expanded it like this: $(x-3)(x-3) = x^2 - 3x - 3x + 9$. We can then combine the like terms: $-3x$ and $-3x$, which gives us $-6x$. So, the expanded form of $(x-3)^2$ is $x^2 - 6x + 9$. Got it? Great! Now, our original expression $2(x-3)^2+5(x-3)-3$ becomes $2(x^2 - 6x + 9) + 5(x-3) - 3$. We're making progress, aren't we? It's all about breaking it down into smaller, manageable steps.

We're now one step closer to getting the polynomial into standard form. Remember, the goal is to transform the original expression into a simplified polynomial in standard form. We're methodically working towards that goal, and the first step was to address the squared term. By expanding $(x-3)^2$, we've simplified one part of the equation, making the overall structure of the expression clearer. It's like dismantling a complicated machine and rebuilding it piece by piece – each step brings us closer to the final product. So keep up the great work, and don't hesitate to pause and review if you need to. Now, let's move on to the next step, where we will simplify the terms further and get closer to our goal of standard form.

Step 2: Distribute and Simplify

Alright, next up we need to distribute the '2' across the terms inside the parentheses in $2(x^2 - 6x + 9)$. This means multiplying each term inside the parentheses by 2. So, $2 * x^2 = 2x^2$, $2 * -6x = -12x$, and $2 * 9 = 18$. This gives us $2x^2 - 12x + 18$. Now, let's not forget about the second term, $5(x-3)$. We need to distribute the '5' as well: $5 * x = 5x$ and $5 * -3 = -15$. Thus, our expression now looks like this: $2x^2 - 12x + 18 + 5x - 15 - 3$. We're doing great! It's all about systematically applying the rules of algebra. Don't worry if you need to go back and re-read a step – understanding is key. Now, we've got a string of terms, and our next job is to combine the like terms. This means combining the terms that have the same variable and exponent (or just the constants, which are the numbers without any variables).

Combining like terms is like organizing things in your closet – put all the shirts together, all the pants together, etc. In our expression $2x^2 - 12x + 18 + 5x - 15 - 3$, the like terms are: the $x$ terms (-12x and 5x) and the constant terms (18, -15, and -3). Combining the x terms: $-12x + 5x = -7x$. Combining the constants: $18 - 15 - 3 = 0$. So, now our expression becomes $2x^2 - 7x + 0$. Simplifying further, we can just write it as $2x^2 - 7x$. Boom! We've simplified the expression into its standard form. From the initial equation, through expansion and distribution, we have now arrived at a simplified polynomial. Remember, practice makes perfect, and with each problem you solve, you'll become more confident and skilled. Now, let's take a final look at our result.

Step 3: Combine Like Terms and Final Answer

Okay, guys, we are at the final stretch! After expanding the squared term and distributing the constants, we're left with $2x^2 - 12x + 18 + 5x - 15 - 3$. As we discussed in Step 2, our task now is to combine like terms. Let's do a recap. The like terms are the ones that have the same variable raised to the same power. So, we'll combine the 'x' terms and the constant terms separately. Our 'x' terms are -12x and +5x. Combining these, we get $-12x + 5x = -7x$. Now, let's tackle the constant terms: 18, -15, and -3. Adding these up: $18 - 15 - 3 = 0$. So, our simplified expression is now $2x^2 - 7x + 0$, which simplifies to $2x^2 - 7x$. And there you have it! The simplified polynomial in standard form is $2x^2 - 7x$. High five! We've successfully transformed our original expression into a much simpler, more manageable form. This final form is in standard form, with the terms arranged in descending order of their exponents, making it easy to identify the degree of the polynomial and its coefficients. You've now seen how to simplify a polynomial step-by-step.

The Final Answer

So, the simplified polynomial in standard form for $2(x-3)^2+5(x-3)-3$ is $\boxed{2x^2 - 7x}$. Awesome work, everyone! You've successfully navigated the process of simplifying a polynomial. Remember, practice makes perfect. The more you work through these types of problems, the easier and more intuitive they will become. Keep up the great work, and don't hesitate to revisit these steps anytime you need a refresher. Now, go forth and conquer those polynomial expressions! And if you liked this tutorial, let us know in the comments! What other math topics would you like us to cover? We are here to help you understand better.