Find The Product Of (4z^2 + 7z - 8) And (-z + 3)

In the realm of algebra, multiplying polynomials is a fundamental skill. This article will guide you through the process of finding the product of two specific polynomials: $(4z^2 + 7z - 8)$ and $(-z + 3)$. We will break down each step, ensuring a clear understanding of the underlying principles. By mastering this technique, you will be well-equipped to tackle more complex algebraic expressions. Let's embark on this journey of algebraic exploration!

Understanding Polynomial Multiplication

Before diving into the specific problem, let's solidify our understanding of polynomial multiplication. At its core, this process involves applying the distributive property. This property states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. When dealing with polynomials, we extend this concept to include multiple terms. Essentially, each term in the first polynomial must be multiplied by each term in the second polynomial. The distributive property is the cornerstone of polynomial multiplication. It allows us to systematically expand the product of two polynomials by ensuring that every term in one polynomial is multiplied by every term in the other. This process is crucial for simplifying algebraic expressions and solving equations. By understanding and applying the distributive property correctly, we can accurately find the product of any two polynomials, regardless of their complexity. The distributive property not only simplifies the multiplication process but also provides a clear and organized approach, reducing the likelihood of errors. Mastering this property is essential for success in algebra and related mathematical fields.

The Distributive Property in Action

To illustrate, consider the product of two binomials: (a + b) and (c + d). Applying the distributive property, we get:

(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd

This simple example highlights the essence of polynomial multiplication. Each term in the first binomial (a and b) is multiplied by each term in the second binomial (c and d). This systematic approach ensures that no term is missed, leading to the correct product. The distributive property is not just a formula; it's a fundamental principle that guides our steps in polynomial multiplication. It allows us to break down complex expressions into manageable parts, making the process more intuitive and less prone to errors. By consistently applying the distributive property, we can confidently navigate polynomial multiplication and achieve accurate results. This understanding is crucial for further algebraic manipulations and problem-solving.

Organizing the Multiplication

For larger polynomials, organizing the multiplication process is crucial. A common method is to use a vertical format, similar to long multiplication with numbers. This helps keep track of terms and ensures that like terms are aligned for easy addition later. Organizing the multiplication process is not just about neatness; it's about efficiency and accuracy. When dealing with polynomials with multiple terms, a structured approach can significantly reduce the chances of making mistakes. The vertical format, for instance, provides a visual aid that helps us keep track of which terms have been multiplied and which haven't. This systematic method ensures that every term in one polynomial is multiplied by every term in the other, leading to a complete and accurate product. Furthermore, aligning like terms in columns makes the final addition step much simpler and less error-prone. By adopting a well-organized approach, we can tackle complex polynomial multiplications with confidence and achieve reliable results. This skill is invaluable in algebra and other areas of mathematics.

Step-by-Step Solution for (4z^2 + 7z - 8) * (-z + 3)

Now, let's apply this knowledge to the given problem: finding the product of $(4z^2 + 7z - 8)$ and $(-z + 3)$. We will follow a step-by-step approach to ensure clarity and accuracy. This methodical approach is key to mastering polynomial multiplication. By breaking down the problem into smaller, manageable steps, we can avoid errors and gain a deeper understanding of the process. Each step builds upon the previous one, leading us to the final solution in a clear and logical manner. This step-by-step method is not only effective for solving this particular problem but also provides a framework for tackling any polynomial multiplication problem. By consistently applying this approach, we can develop proficiency and confidence in our algebraic skills.

Step 1: Distribute -z

First, we distribute the term -z across the first polynomial:

-z * (4z^2 + 7z - 8) = -4z^3 - 7z^2 + 8z

The first step in polynomial multiplication involves distributing one of the terms across the other polynomial. In this case, we begin by distributing -z across the polynomial $(4z^2 + 7z - 8)$. This means we multiply -z by each term within the parentheses. This process is a direct application of the distributive property, which is the cornerstone of polynomial multiplication. By carefully multiplying -z by each term, we ensure that no term is missed and that the resulting expression is accurate. This initial step sets the foundation for the rest of the solution, so it's crucial to perform it meticulously. A clear understanding of this distribution process is essential for successfully multiplying polynomials of any size or complexity.

Step 2: Distribute 3

Next, we distribute the term 3 across the first polynomial:

3 * (4z^2 + 7z - 8) = 12z^2 + 21z - 24

Following the distribution of -z, we now distribute the term 3 across the same polynomial, $(4z^2 + 7z - 8)$. This step mirrors the previous one, applying the distributive property to multiply 3 by each term within the parentheses. Again, meticulous attention to detail is crucial to ensure the accuracy of the resulting expression. By carefully performing this distribution, we obtain another set of terms that will contribute to the final product. This step, combined with the previous one, covers all the necessary multiplications to expand the expression. Understanding the consistent application of the distributive property is key to mastering polynomial multiplication and achieving accurate results.

Step 3: Combine Like Terms

Now, we add the results from Step 1 and Step 2:

(-4z^3 - 7z^2 + 8z) + (12z^2 + 21z - 24) = -4z^3 + 5z^2 + 29z - 24

The final step in polynomial multiplication involves combining like terms. This means identifying terms with the same variable and exponent and adding their coefficients. In this case, we combine the terms resulting from the distributions in the previous steps. This process simplifies the expression and presents the final product in its most concise form. Careful attention to detail is essential to ensure that like terms are correctly identified and combined. This step not only completes the multiplication process but also demonstrates the importance of algebraic simplification. By combining like terms, we obtain a clear and understandable representation of the product of the two polynomials.

The Final Result

Therefore, the product of $(4z^2 + 7z - 8)$ and $(-z + 3)$ is $-4z^3 + 5z^2 + 29z - 24$.

In conclusion, multiplying polynomials requires a systematic approach, primarily utilizing the distributive property. By breaking down the process into manageable steps, we can accurately find the product of any two polynomials. This skill is fundamental in algebra and serves as a building block for more advanced mathematical concepts.

Identifying the Missing Coefficients

The original problem presented the product in the form -4z^3 + oxed{ ext{ }} z^2 + oxed{ ext{ }} z - 24. By comparing this with our result, we can identify the missing coefficients.

Filling in the Blanks

From our solution, we found the product to be $-4z^3 + 5z^2 + 29z - 24$. Therefore, the missing coefficients are 5 and 29.

Practice Problems

To further solidify your understanding, try these practice problems:

1. Find the product of $(2x + 3)$ and $(x - 4)$.
2. Find the product of $(3y^2 - y + 2)$ and $(2y + 1)$.
3. Find the product of $(a + b)$ and $(a - b)$.

By working through these practice problems, you can reinforce your understanding of polynomial multiplication and develop your skills in algebraic manipulation. Practice is key to mastering any mathematical concept, and these problems provide an opportunity to apply the techniques discussed in this article. As you solve these problems, pay close attention to the steps involved and ensure that you are applying the distributive property correctly. The more you practice, the more confident and proficient you will become in multiplying polynomials.

Conclusion

Mastering polynomial multiplication is crucial for success in algebra and beyond. By understanding the distributive property and following a systematic approach, you can confidently tackle any polynomial multiplication problem. Remember to practice regularly to hone your skills and build your algebraic foundation.