Simplify Polynomial Expression: AB-C^2

Hey math whizzes! Ever find yourself staring at a bunch of polynomials and wondering how to make them, like, way simpler? Well, you've come to the right place, guys. Today, we're diving deep into a classic algebra problem: simplifying the expression $AB - C^2$ where $A$, $B$, and $C$ are specific polynomials. We're talking about $A = x^2$, $B = 3x + 2$, and $C = x - 3$. This isn't just about crunching numbers; it's about understanding how to manipulate algebraic expressions, a fundamental skill in mathematics that opens doors to solving more complex problems down the line. Think of it as building blocks – the more solid your foundation in simplifying expressions, the higher you can build your mathematical structures. We'll break down each step, ensuring you understand not just what to do, but why you're doing it. So grab your calculators (or just your brains!), and let's get this polynomial party started!

Understanding the Polynomials: A, B, and C

Before we jump into the main event, let's get reacquainted with our players: $A$, $B$, and $C$. We've got $A = x^2$. This is a simple quadratic monomial, meaning it's a single term with a variable raised to the power of two. Then there's $B = 3x + 2$. This is a linear binomial, composed of two terms: $3x$ (a term with a variable to the first power) and $2$ (a constant term). Finally, we have $C = x - 3$. This is also a linear binomial, similar to $B$, with an $x$ term and a constant term. Understanding the structure of these polynomials is key because it dictates how we'll multiply and square them. For instance, multiplying $A$ by $B$ will involve distributing $x^2$ to each term in $3x+2$, while squaring $C$ will require us to multiply $(x-3)$ by itself. These basic operations, multiplication and squaring, are the building blocks for tackling the entire expression $AB - C^2$. We'll be using the distributive property and the FOIL method (or simply multiplying each term by each other) extensively, so a solid grasp of these fundamental algebraic techniques is super important. It’s like learning your scales before playing a concerto – mastering these simple steps makes the whole piece flow beautifully. We're not just substituting values here; we're performing operations that change the form of the expressions, moving us closer to a simplified, elegant answer. This is where the magic of algebra really shines, transforming complex-looking expressions into manageable ones.

Step 1: Calculate AB

Alright team, the first mission in simplifying $AB - C^2$ is to calculate the product of $A$ and $B$. Remember, we have $A = x^2$ and $B = 3x + 2$. To find $AB$, we need to multiply $x^2$ by the entire polynomial $3x + 2$. This is a straightforward application of the distributive property. We take the term outside the parenthesis ($x^2$) and multiply it by each term inside the parenthesis ($3x$ and $2$).

So, we have:

$AB = x^2 imes (3x + 2)$

Let's distribute:

$AB = (x^2 imes 3x) + (x^2 imes 2)$

Now, let's simplify each part. For the first part, $x^2 imes 3x$, we multiply the coefficients (which are $1$ and $3$) and add the exponents of the variable $x$. Remember, $x$ is the same as $x^1$. So, $1 imes 3 = 3$, and $x^2 imes x^1 = x^{2+1} = x^3$. This gives us $3x^3$.

For the second part, $x^2 imes 2$, it's a bit simpler. We just multiply the coefficient $1$ from $x^2$ by the constant $2$, which gives us $2$. The $x^2$ term remains as it is since there's no other $x$ term to combine it with. So, this part becomes $2x^2$.

Putting it all together, we get:

$AB = 3x^3 + 2x^2$

Boom! That's the first half of our puzzle solved. We've successfully multiplied $A$ and $B$. This result, $3x^3 + 2x^2$, is the simplified form of the $AB$ portion of our expression. It's crucial to get this step right because any errors here will carry over to the final answer. Always double-check your exponent rules when multiplying variables – adding the powers is the name of the game! This step demonstrates how a monomial can be distributed across a binomial, transforming a multiplication problem into a sum of terms. It's a foundational skill, and seeing $3x^3 + 2x^2$ emerge is the first win in our algebraic journey today. We're building momentum, guys!

Step 2: Calculate C²

Now, let's tackle the second part of our expression: $C^2$. We know that $C = x - 3$. Squaring $C$ means multiplying the polynomial $C$ by itself. So, we need to calculate $(x - 3) imes (x - 3)$.

This is where we often use the FOIL method (First, Outer, Inner, Last) or simply apply the distributive property twice. Let's go with the distributive approach, which is essentially what FOIL is.

We take the first term of the first parenthesis, $x$, and multiply it by each term in the second parenthesis ($(x - 3)$):

$x imes (x - 3) = (x imes x) + (x imes -3) = x^2 - 3x$

Next, we take the second term of the first parenthesis, $-3$, and multiply it by each term in the second parenthesis ($(x - 3)$):

$-3 imes (x - 3) = (-3 imes x) + (-3 imes -3) = -3x + 9$

Now, we add the results from both distributions:

$C^2 = (x^2 - 3x) + (-3x + 9)$

Finally, we combine like terms. The like terms here are $-3x$ and $-3x$. Combining them gives us $-3x - 3x = -6x$.

So, the simplified form of $C^2$ is:

$C^2 = x^2 - 6x + 9$

Alternatively, you might remember the formula for squaring a binomial: $(a - b)^2 = a^2 - 2ab + b^2$. In our case, $a = x$ and $b = 3$. Plugging these into the formula gives us:

$(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$

See? Same result! This confirms our calculation. Squaring a binomial is a common operation, and recognizing patterns like this can save you time and reduce errors. It’s essential to pay close attention to the signs, especially when multiplying negative numbers. The $-3$ multiplied by $-3$ correctly yields a positive $9$. This step is crucial as it involves squaring a binomial, which often trips people up with the middle term. Getting this right means we're one step closer to the final answer, and we've successfully squared our polynomial $C$. High five!

Step 3: Calculate AB - C²

Now for the grand finale, guys! We've successfully calculated $AB$ and $C^2$. Our task is to find $AB - C^2$. We simply substitute the simplified expressions we found in the previous steps.

We have:

$AB = 3x^3 + 2x^2$

And:

$C^2 = x^2 - 6x + 9$

So, we need to calculate:

$(3x^3 + 2x^2) - (x^2 - 6x + 9)$

This is the most critical step for potential errors, as we are subtracting an entire polynomial. Remember that when you subtract a polynomial, you must distribute the negative sign to each term inside the parentheses.

Let's rewrite the expression with the distributed negative sign:

$3x^3 + 2x^2 - x^2 - (-6x) - (+9)$

Simplifying the signs:

$3x^3 + 2x^2 - x^2 + 6x - 9$

Now, the final part is to combine like terms. We look for terms that have the same variable raised to the same power.

• The $x^3$ term is $3x^3$. There are no other $x^3$ terms, so it stays as is.
• The $x^2$ terms are $2x^2$ and $-x^2$. Combining them gives us $2x^2 - x^2 = 1x^2$, or simply $x^2$.
• The $x$ term is $6x$. There are no other $x$ terms, so it remains $6x$.
• The constant term is $-9$. There are no other constant terms, so it stays $-9$.

Putting all the combined terms together in descending order of their exponents, we get our final simplified expression:

$3x^3 + x^2 + 6x - 9$

This is the result of $AB - C^2$ in its simplest form. Take a moment to appreciate how we went from a complex expression involving multiplication and subtraction of polynomials to a single, neat polynomial. This is the power of algebraic manipulation! Always be super careful with signs when subtracting polynomials; that's where most mistakes happen. Make sure you distribute that negative sign to every single term within the subtrahend. Well done, everyone!

Conclusion: The Simplified Expression

So, after all that hard work, guys, we've arrived at the simplest form of the expression $AB - C^2$. Let's recap our journey. We started with $A = x^2$, $B = 3x + 2$, and $C = x - 3$. Our first step was to calculate $AB$, which we found to be $3x^3 + 2x^2$ by using the distributive property. Next, we squared $C$, calculating $(x - 3)^2$ to get $x^2 - 6x + 9$, employing either the FOIL method or the binomial square formula. The final and most crucial step was subtracting the squared $C$ from $AB$. This involved carefully distributing the negative sign to each term in $C^2$ before combining all the like terms. The subtraction yielded $(3x^3 + 2x^2) - (x^2 - 6x + 9)$, which simplified to $3x^3 + 2x^2 - x^2 + 6x - 9$. Finally, combining the $x^2$ terms ($2x^2 - x^2 = x^2$) gave us our ultimate answer: $3x^3 + x^2 + 6x - 9$.

This result, $3x^3 + x^2 + 6x - 9$, is the answer in its simplest form. It's a cubic polynomial, meaning the highest power of $x$ is 3. When comparing this to the multiple-choice options provided (A, B, C, D), we can see that our calculated result matches option D. It's incredibly satisfying to see the pieces fall into place and arrive at the correct solution through careful application of algebraic rules. Remember, practice is key! The more you work with polynomials, the more comfortable you'll become with these operations, and the faster you'll be able to spot patterns and potential pitfalls. Keep practicing, keep exploring, and you'll master these algebraic challenges in no time. You guys are awesome!