Mastering Polynomial Multiplication: A Step-by-Step Guide

Welcome to our comprehensive guide on mastering polynomial multiplication! In the realm of mathematics, understanding how to effectively multiply polynomials is a fundamental skill that opens doors to solving more complex algebraic problems. Today, we're going to delve deep into the process, focusing on a specific example: multiplying $-y$ by the polynomial $4x^3 - 5x^2y + xy^2 + 3y^3$. This might look a little intimidating at first glance, but with a clear, step-by-step approach, you'll find it quite manageable. We'll break down each component, explaining the distributive property and the rules of exponents, ensuring you not only get the right answer but also understand why it's the right answer. Get ready to boost your algebraic confidence as we embark on this mathematical journey together.

Understanding the Core Principle: The Distributive Property

The essence of multiplying a monomial (a single term) by a polynomial (an expression with multiple terms) lies in the distributive property. This property is a cornerstone of algebra and states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products. In our case, the number outside the parentheses is $-y$, and the polynomial inside is $4x^3 - 5x^2y + xy^2 + 3y^3$. So, we need to distribute the $-y$ to each term within the parentheses. This means we will perform the multiplication operation four times: once for $4x^3$, once for $-5x^2y$, once for $xy^2$, and finally, once for $3y^3$. It's crucial to remember that when we distribute, we are essentially applying the multiplication to every single part of the expression within the parentheses. Think of it like dealing cards to every player at the table; each term in the polynomial is a player, and $-y$ is the deck of cards being dealt to each one. This systematic approach ensures that no term is left out and that the entire expression is correctly transformed. By consistently applying this distributive property, we can systematically break down complex multiplications into a series of simpler ones, making the overall process much more approachable and less prone to errors. It’s the fundamental rule that governs how these algebraic expressions interact and transform, so a solid grasp of it is paramount for success in algebra.

Step 1: Distribute to the First Term

Let's begin by multiplying $-y$ with the first term inside the parentheses, which is $4x^3$. To do this, we multiply the coefficients and the variables separately, keeping in mind the rules of exponents. The coefficient of $-y$ is $-1$, and the coefficient of $4x^3$ is $4$. Multiplying these gives us $(-1) imes 4 = -4$. For the variables, we have $y$ and $x^3$. Since they are different variables, we simply write them next to each other. However, we must also consider the order. It's conventional to write variables in alphabetical order. So, we have $x^3$ and then $y$. Thus, the product of $-y$ and $4x^3$ is $-4x^3y$. This initial step sets the stage for the rest of the multiplication. It's important to be meticulous with the signs. The negative sign in $-y$ will affect the sign of the resulting term. Since we are multiplying a negative by a positive, the result is negative. Furthermore, when combining variables, we adhere to the rule that $y imes ( ext{no } y ext{ term}) = y^1$. If there were multiple powers of the same variable, we would add their exponents (e.g., $y^2 imes y^3 = y^{2+3} = y^5$). In this specific case, since $x^3$ doesn't have a $y$ term and $y$ doesn't have an $x$ term, we just combine them as $x^3y$. This careful attention to detail in signs and variable powers is key to accuracy in polynomial multiplication. We are essentially applying the rule $(ab)(cd) = (ac)(bd)$ where $a = -1$, $b = y$, $c = 4$, and $d = x^3$, and then arranging the variables alphabetically. The product is $-4x^3y$. This is the first piece of our final answer, and it's crucial to get this part right before moving on to the next term.

Step 2: Distribute to the Second Term

Next, we take $-y$ and multiply it by the second term in the polynomial, which is $-5x^2y$. Again, we multiply the coefficients and the variables. The coefficients are $-1$ (from $-y$) and $-5$. Multiplying these gives us $(-1) imes (-5) = 5$. Now for the variables: we have $y$ from the monomial and $x^2y$ from the polynomial. When we multiply $y$ by $y$, we are multiplying $y^1$ by $y^1$. Using the rule of exponents that states when multiplying powers with the same base, we add the exponents, we get $y^{1+1} = y^2$. The variable $x^2$ remains as it is. Combining these parts and arranging the variables alphabetically, we get $x^2y^2$. Therefore, the product of $-y$ and $-5x^2y$ is $5x^2y^2$. Notice how the multiplication of two negative numbers ($-1$ and $-5$) resulted in a positive number ($5$). This sign change is a critical aspect of polynomial multiplication. It's easy to make mistakes with signs, so double-checking each multiplication step is highly recommended. The rule $y imes y = y^2$ is a direct application of the product of powers property, $a^m imes a^n = a^{m+n}$. Here, $a=y$, $m=1$, and $n=1$. So, $y^1 imes y^1 = y^{1+1} = y^2$. When combining this with the $x^2$ term, we maintain alphabetical order, resulting in $x^2y^2$. This term, $5x^2y^2$, is the second part of our expanded polynomial. Each successful distribution brings us closer to the final, simplified expression, building confidence with every correctly calculated term.

Step 3: Distribute to the Third Term

Moving on, we now multiply $-y$ by the third term in the parentheses, which is $xy^2$. Let's break it down. The coefficient of $-y$ is $-1$, and the coefficient of $xy^2$ is $1$ (since $x$ is the same as $1x$). So, $(-1) imes 1 = -1$. For the variables, we have $y$ from the monomial and $xy^2$ from the polynomial. We need to multiply $y$ by $y^2$. This is $y^1 imes y^2$. Using the exponent rule, we add the exponents: $1 + 2 = 3$. So, $y^1 imes y^2 = y^3$. The variable $x$ from $xy^2$ has no corresponding $x$ in $-y$, so it remains as $x$. Combining these parts and arranging alphabetically, we get $xy^3$. Thus, the product of $-y$ and $xy^2$ is $-xy^3$. In this step, we multiplied a negative coefficient ($-1$) by a positive coefficient ($1$), resulting in a negative product. This reinforces the importance of paying close attention to signs. The variable $x$ is simply carried over as there's no $x$ term in $-y$ to combine with it. The multiplication of $y$ and $y^2$ exemplifies the power rule again: $y^1 imes y^2 = y^{1+2} = y^3$. This term, $-xy^3$, is the third component of our expanded polynomial. Each step, though seemingly small, contributes significantly to the overall accuracy of the final result. By breaking down the problem, we minimize the chances of error and build a solid understanding of the process.

Step 4: Distribute to the Fourth Term

Finally, we reach the last term inside the parentheses: $3y^3$. We multiply $-y$ by $3y^3$. The coefficients are $-1$ (from $-y$) and $3$. Multiplying them gives us $(-1) imes 3 = -3$. Now, let's deal with the variables. We have $y$ from the monomial and $y^3$ from the polynomial. Multiplying $y$ by $y^3$ is $y^1 imes y^3$. Applying the exponent rule, we add the exponents: $1 + 3 = 4$. So, $y^1 imes y^3 = y^4$. Therefore, the product of $-y$ and $3y^3$ is $-3y^4$. This is the final term we obtain from our distributions. We multiplied a negative coefficient by a positive coefficient, resulting in a negative product. The variables $y$ combine according to the rule of exponents, increasing the power of $y$ to $4$. This step completes the distribution process. It's satisfying to see how each term in the original polynomial is transformed by the multiplication with $-y$. The final result will be the sum of all these individual products. This final product, $-3y^4$, is the last piece of our expanded polynomial. We have now successfully multiplied $-y$ by every term within the parentheses, addressing all parts of the expression.

Combining the Results

Now that we have successfully distributed $-y$ to each term within the polynomial $4x^3 - 5x^2y + xy^2 + 3y^3$, we need to combine all the resulting terms to form our final answer. We found the following products:

• From Step 1: $-4x^3y$
• From Step 2: $5x^2y^2$
• From Step 3: $-xy^3$
• From Step 4: $-3y^4

Since all these terms have different variable combinations (e.g., $x^3y$, $x^2y^2$, $xy^3$, $y^4$), they are unlike terms. Unlike terms cannot be combined further through addition or subtraction. Therefore, the final answer is simply the collection of these individual products, written in order. So, the result of multiplying $-y extbf{(}4x^3 - 5x^2y + xy^2 + 3y^3 extbf{)}$ is $-4x^3y + 5x^2y^2 - xy^3 - 3y^4$. It's important to present the final answer clearly, ensuring all signs are correct and the terms are listed distinctly. This expression represents the fully expanded form of the original multiplication problem. Each term in the final expression corresponds directly to one of the distribution steps, confirming that our process was thorough and accurate. The absence of like terms means no further simplification is possible, and this is our definitive solution. This process highlights the power of the distributive property and the rules of exponents in transforming and simplifying algebraic expressions. It's a critical skill for progressing in algebra and tackling more advanced mathematical concepts.

Conclusion: Your Polynomial Multiplication Mastery

We've successfully navigated the process of multiplying a monomial by a polynomial, using the example $-y(4x^3 - 5x^2y + xy^2 + 3y^3)$. By systematically applying the distributive property and the rules of exponents, we've transformed a complex expression into a simpler, expanded form: $-4x^3y + 5x^2y^2 - xy^3 - 3y^4$. Remember, the key steps involve multiplying the coefficients, combining the variables, and paying close attention to the signs at each stage. Practice is essential for mastering any mathematical skill, and polynomial multiplication is no exception. The more you practice, the more intuitive these steps will become, and the faster you'll be able to solve these problems accurately. Don't be discouraged if you make mistakes; they are a natural part of the learning process. Analyze your errors, understand where you went wrong, and keep practicing. With dedication, you'll soon find yourself confidently multiplying polynomials and tackling even more challenging algebraic tasks. Keep exploring the fascinating world of mathematics!

For further exploration and practice on algebraic concepts, you can visit Khan Academy, a fantastic resource for learning mathematics from basic to advanced levels.