Binomial By Trinomial Multiplication Explained

Hey guys! Today, we're diving into multiplying a binomial by a trinomial. Specifically, we'll break down the multiplication of $(3x + y)$ by $(x^2 + 2y + 4)$. This might sound intimidating, but with a structured approach, it’s super manageable. We'll use a table to keep everything organized and clear. So, let's get started and make sure you understand every step! This is a fundamental concept in algebra, and mastering it will definitely help you in more advanced topics. Ready? Let's jump right in!

Understanding the Setup

Before we begin, let's quickly recap what binomials and trinomials are. A binomial is a polynomial with two terms, like our $(3x + y)$. A trinomial, on the other hand, is a polynomial with three terms, such as $(x^2 + 2y + 4)$. Multiplying these together means each term in the binomial needs to be multiplied by each term in the trinomial. Organization is key here, and that's where our table comes in handy. Think of it as a grid that helps us keep track of each multiplication, ensuring we don’t miss anything. It's like a cheat sheet that prevents silly mistakes! Plus, understanding this process will make more complex polynomial multiplications much easier down the road. So, pay close attention, and let's make sure you've got this down pat.

The Multiplication Table

Here’s the multiplication table provided, which we'll dissect step by step:

Breaking Down the Table

Now, let's break down how each entry in the table is derived. This table represents the distributive property in action, ensuring every term in the first polynomial is multiplied by every term in the second polynomial. This structured approach minimizes errors and clarifies the process. Each cell in the table represents the product of the corresponding terms from the binomial and trinomial. By filling out each cell correctly, we systematically expand the expression and set ourselves up for combining like terms to simplify the result. Understanding the logic behind each multiplication will reinforce your understanding and allow you to apply this method to other polynomial multiplications. So, let's go through each cell to solidify your knowledge.

Row 1: Multiplying $3x$

• $3x * x^2 = 3x^3$: When you multiply $3x$ by $x^2$, you're essentially multiplying the coefficients (3 and 1) and adding the exponents of $x$ (1 and 2). Thus, $3 * 1 = 3$ and $x^1 * x^2 = x^{1+2} = x^3$. Combining these gives us $3x^3$. This is a straightforward application of the power rule in exponents. Remember, when multiplying terms with the same base, you add their exponents. Keep this rule in mind as we continue through the table.
• $3x * 2y = 6xy$: Here, we multiply $3x$ by $2y$. Multiply the coefficients (3 and 2) to get 6. Since the variables are different ($x$ and $y$), we simply write them next to each other. So, $3 * 2 = 6$, and combining the variables gives us $6xy$. Make sure you keep track of the variables and their order. Typically, we list them alphabetically, but it doesn't change the value if you write $yx$.
• $3x * 4 = 12x$: Multiplying $3x$ by 4 is simple. Just multiply the coefficient 3 by 4 to get 12, and keep the $x$. So, $3 * 4 = 12$, and we have $12x$. This is a basic multiplication, but it's essential to get it right. Always double-check your arithmetic to avoid mistakes.

Row 2: Multiplying $y$

• $y * x^2 = x^2y$: When you multiply $y$ by $x^2$, you simply write the terms next to each other. It's standard practice to write the variables in alphabetical order, so we write $x^2y$. The multiplication is straightforward, but maintaining consistent notation helps in later steps. Remember, the order of multiplication doesn't affect the result, but keeping the variables in alphabetical order makes it easier to combine like terms later.
• $y * 2y = 2y^2$: Here, we multiply $y$ by $2y$. Multiply the coefficients (1 and 2) to get 2. Since we are multiplying $y$ by $y$, we add the exponents: $y^1 * y^1 = y^{1+1} = y^2$. Combining these gives us $2y^2$. This is another application of the power rule. Make sure you understand how the exponents are added when multiplying terms with the same base.
• $y * 4 = 4y$: Multiplying $y$ by 4 is straightforward. Multiply the coefficient 1 (understood to be in front of the y) by 4 to get 4, and keep the $y$. So, $1 * 4 = 4$, and we have $4y$. This is a simple multiplication, but accuracy is key. Double-check to ensure you haven't made any errors.

Combining the Terms

Now that we have all the individual products from the table, we need to combine them to get the expanded form of the original expression. This involves adding all the terms from the table together. Be careful to correctly identify and combine like terms to simplify the expression as much as possible. This is where attention to detail really pays off. Make sure you have a good grasp of what constitutes a like term (same variables raised to the same powers) before proceeding.

Assembling the Expanded Form

From the table, we have the following terms:

$3x^3 + 6xy + 12x + x^2y + 2y^2 + 4y$

Checking for Like Terms

In this expanded form, we look for like terms, which are terms that have the same variables raised to the same powers. In our case, there are no like terms to combine. Each term has a unique combination of variables and exponents.

Final Expanded Form

Since there are no like terms to combine, our final expanded form is:

$3x^3 + 6xy + 12x + x^2y + 2y^2 + 4y$

Conclusion

Alright, guys, we've successfully multiplied the binomial $(3x + y)$ by the trinomial $(x^2 + 2y + 4)$ using a structured table method. We broke down each multiplication step by step, combined the terms, and arrived at the final expanded form: $3x^3 + 6xy + 12x + x^2y + 2y^2 + 4y$. This method not only helps in keeping track of the terms but also provides a clear understanding of the distributive property in action. Remember, practice makes perfect, so keep working on similar problems to solidify your understanding. You've got this!