Multiplying Polynomials: Which Table Shows The Correct Solution?

Understanding polynomial multiplication is a crucial skill in algebra. This article will guide you through the process of multiplying two polynomials, $x^2 - 4x + 4$ and $6x + 3$, and help you identify the correct table that represents the solution. Polynomial multiplication might seem daunting at first, but with a systematic approach, it becomes manageable. We'll break down each step and explain the underlying principles, ensuring you grasp the concept thoroughly. This is more than just finding the right answer; it's about building a strong foundation in algebraic manipulation.

Breaking Down Polynomial Multiplication

When tackling polynomial multiplication, the distributive property is your best friend. This property dictates that each term in the first polynomial must be multiplied by each term in the second polynomial. It's like shaking hands at a party – everyone needs to greet everyone else! Let's consider our two polynomials: $x^2 - 4x + 4$ and $6x + 3$. The first polynomial has three terms ($x^2$, $-4x$, and $4$), and the second has two terms ($6x$ and $3$). So, we'll have a total of 3 * 2 = 6 individual multiplications to perform. This is where organizing your work becomes essential to avoid errors. Using a table, as suggested in the original question, is an excellent way to keep track of these multiplications. Each cell in the table will represent the product of a term from the first polynomial and a term from the second polynomial. The table method provides a visual structure that can significantly reduce the chances of making mistakes, especially when dealing with larger polynomials. In the following sections, we'll demonstrate how to set up and fill out this table effectively.

Setting Up the Multiplication Table

The multiplication table acts as a visual aid, neatly organizing the multiplication process. Think of it as a grid where each cell represents the product of two terms. To set it up, list the terms of the first polynomial ($x^2 - 4x + 4$) across the top as column headers and the terms of the second polynomial ($6x + 3$) along the left side as row headers. This creates a framework where each cell corresponds to a specific multiplication. For example, the cell in the top-left corner will represent the product of $x^2$ and $6x$. The cell next to it will represent the product of $-4x$ and $6x$, and so on. This structured approach is crucial for ensuring that every term is multiplied correctly and no terms are missed. A well-organized table minimizes the risk of errors and makes it easier to identify and combine like terms later on. By visualizing the multiplication process in this way, you can gain a clearer understanding of how the distributive property is applied in polynomial multiplication.

Performing the Multiplication

Now, let's fill in the table by performing the multiplication. In each cell, multiply the corresponding terms from the row and column headers. For instance, in the first cell (top-left), we multiply $x^2$ by $6x$, resulting in $6x^3$. In the next cell, we multiply $-4x$ by $6x$, which gives us $-24x^2$. Continuing this process for each cell, we methodically calculate the products. Accuracy is paramount at this stage; a single error in multiplication can throw off the entire solution. Pay close attention to the signs (positive or negative) and the exponents. Remember the rules of exponents: when multiplying terms with the same base, you add the exponents. For example, $x^2 * x = x^(2+1) = x^3$. This step-by-step multiplication ensures that we account for every possible combination of terms, laying the groundwork for the next stage: combining like terms. By carefully performing each multiplication and double-checking your work, you can build confidence in your final answer.

Combining Like Terms

After filling the multiplication table, the next step is to combine like terms. Like terms are those that have the same variable raised to the same power. In our expanded polynomial, these terms are scattered throughout the table, and combining them simplifies the expression. For example, we might have terms like $-24x^2$ and $3x^2$. These are like terms because they both have the variable x raised to the power of 2. To combine them, we simply add their coefficients: -24 + 3 = -21. So, these two terms combine to form $-21x^2$. Similarly, we would look for other terms with the same variable and exponent and add their coefficients. This process of combining like terms is essential for simplifying the polynomial and presenting it in its most concise form. It also helps to reduce the complexity of the expression, making it easier to work with in subsequent calculations or applications. Think of it as tidying up after the multiplication, grouping similar items together to make the final result clearer and more manageable.

Identifying the Correct Table

Now, let's compare the table provided in the original question with the multiplication we've performed. The question presents a table and asks which one accurately represents the multiplication of $x^2 - 4x + 4$ and $6x + 3$. We've already established the process of setting up the table, performing the multiplication, and identifying like terms. By comparing the values in each cell of the given table with our calculations, we can determine if it accurately reflects the multiplication process. Look for any discrepancies in the signs, coefficients, or exponents. A single mistake in any cell can invalidate the entire table. This step is crucial for verifying the solution and ensuring that we've correctly applied the distributive property and combined like terms. It's like proofreading a document; you're carefully reviewing each element to catch any errors. By meticulously comparing the provided table with our calculations, we can confidently identify the correct representation of the polynomial multiplication.

Analyzing the Provided Table

Let's take a closer look at the table provided in the original question:

|       | $x^2$     | $-4x$     | $4$      |
| :---- | :-------- | :-------- | :------- |
| $6x$  | $6x^3$    | $-24x^2$   | $24x$    |
| $3$   | $3x^2$    | $-12x$    | $12$     |

We need to verify if the entries in this table are correct. Remember, each cell represents the product of the corresponding row and column headers. Let's examine each cell individually:

• Cell 1 (6x * x^2): 6x * x^2 = 6x^3. This is correct.
• Cell 2 (6x * -4x): 6x * -4x = -24x^2. This is also correct.
• Cell 3 (6x * 4): 6x * 4 = 24x. Correct again.
• Cell 4 (3 * x^2): 3 * x^2 = 3x^2. This is correct as well.
• Cell 5 (3 * -4x): 3 * -4x = -12x. Correct.
• Cell 6 (3 * 4): 3 * 4 = 12. This is also correct.

All the individual multiplications in the table are accurate. This indicates that the table correctly represents the multiplication of the two polynomials.

The Final Result

The table accurately represents the multiplication of the polynomials $x^2 - 4x + 4$ and $6x + 3$. To get the final result, we need to combine the terms from the table:

$6x^3 - 24x^2 + 24x + 3x^2 - 12x + 12$

Now, let's combine the like terms:

• $-24x^2$ and $3x^2$ are like terms. Combining them gives us $-21x^2$.
• $24x$ and $-12x$ are like terms. Combining them gives us $12x$.

So, the simplified polynomial is:

$6x^3 - 21x^2 + 12x + 12$

Therefore, the table correctly shows the intermediate steps of the multiplication, and by combining the terms, we arrive at the final polynomial expression.

Conclusion

In conclusion, the provided table accurately represents the multiplication of the polynomials $x^2 - 4x + 4$ and $6x + 3$. We systematically verified each cell in the table, confirming that the individual multiplications were performed correctly. We then combined like terms to arrive at the simplified polynomial expression: $6x^3 - 21x^2 + 12x + 12$. This exercise highlights the importance of the distributive property and the organization provided by the table method in polynomial multiplication. Mastering these skills is crucial for success in algebra and beyond. Remember, practice makes perfect! The more you work with polynomial multiplication, the more comfortable and confident you'll become. Don't hesitate to tackle similar problems and explore different methods to solidify your understanding. For further exploration of polynomial operations and algebraic concepts, consider visiting trusted educational resources such as Khan Academy. Their comprehensive lessons and practice exercises can help you deepen your knowledge and build a strong foundation in mathematics.