Multiplying Polynomials A Step-by-Step Guide To (2x^2 + 4x - 3)(x^2 - 2x + 5)

In the realm of mathematics, particularly in algebra, multiplying polynomials is a fundamental skill. This process involves distributing each term of one polynomial across every term of the other polynomial, a technique that may seem daunting at first but becomes manageable with a systematic approach. In this comprehensive guide, we will dissect the multiplication of two specific polynomials: $\left(2x^2 + 4x - 3\right)$ and $\left(x^2 - 2x + 5\right)$. This exercise not only reinforces the basic principles of polynomial multiplication but also provides a clear pathway for tackling more complex algebraic expressions. Understanding how to multiply polynomials is crucial for various mathematical applications, including solving equations, simplifying expressions, and even in calculus. The ability to confidently navigate these operations opens doors to more advanced mathematical concepts and problem-solving scenarios. So, let’s embark on this journey of polynomial multiplication, breaking down each step to ensure a solid grasp of the methodology involved.

Understanding Polynomial Multiplication

Before diving into the specifics, it’s essential to understand the underlying principle of polynomial multiplication. At its core, this process relies on the distributive property, which states that each term in one polynomial must be multiplied by each term in the other polynomial. This ensures that no term is left out and that the resulting expression accurately reflects the product of the two polynomials. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The key to multiplying them effectively lies in a systematic approach. This involves organizing terms, paying close attention to signs, and combining like terms after the distribution process is complete. Mastering this process is not just about getting the right answer; it’s about developing a deeper understanding of algebraic structures and how they interact. The distributive property is the workhorse of this operation, allowing us to break down a seemingly complex problem into smaller, manageable steps. Each term's distribution is like a mini-multiplication problem, and by carefully handling each one, we can build up to the complete product of the polynomials. Furthermore, understanding polynomial multiplication is a stepping stone to more advanced topics such as polynomial division and factoring. These operations are interconnected, and proficiency in one area often enhances understanding in others. So, by mastering the basics, we lay a strong foundation for future mathematical endeavors.

Step-by-Step Multiplication of $\left(2x^2 + 4x - 3\right) \left(x^2 - 2x + 5\right)$

Now, let’s apply the distributive property to our specific problem: multiplying $\left(2x^2 + 4x - 3\right)$ by $\left(x^2 - 2x + 5\right)$. This involves multiplying each term of the first polynomial by each term of the second polynomial. We start by multiplying $2x^2$ by each term in the second polynomial, then repeat the process for $4x$ and $-3$. This methodical approach ensures that we account for every possible product, preventing errors and maintaining clarity throughout the process. Remember, each multiplication step involves multiplying the coefficients and adding the exponents of the variables. For instance, when multiplying $2x^2$ by $x^2$, we multiply the coefficients (2 * 1 = 2) and add the exponents (2 + 2 = 4), resulting in $2x^4$. This careful attention to detail is crucial for accurate polynomial multiplication. After distributing each term, we will have a series of terms that need to be simplified. This simplification involves combining like terms, which are terms with the same variable and exponent. For example, $3x^2$ and $-5x^2$ are like terms and can be combined to $-2x^2$. The systematic application of the distributive property, followed by the careful combination of like terms, is the key to successfully multiplying polynomials. This step-by-step process not only provides the correct answer but also builds a strong foundation for handling more complex algebraic manipulations.

1. Distribute $2x^2$

We begin by distributing the first term of the first polynomial, $2x^2$, across the second polynomial $\left(x^2 - 2x + 5\right)$. This means multiplying $2x^2$ by each term within the parentheses: $x^2$, $-2x$, and $5$. Each of these multiplications is a straightforward application of the power rule and coefficient multiplication. When multiplying $2x^2$ by $x^2$, we multiply the coefficients (2 * 1 = 2) and add the exponents of $x$ (2 + 2 = 4), resulting in $2x^4$. Similarly, when multiplying $2x^2$ by $-2x$, we multiply the coefficients (2 * -2 = -4) and add the exponents of $x$ (2 + 1 = 3), giving us $-4x^3$. Finally, multiplying $2x^2$ by 5 involves simply multiplying the coefficient 2 by 5, resulting in $10x^2$. This systematic distribution ensures that we account for each interaction between $2x^2$ and the terms of the second polynomial. The result of this first step is the expression $2x^4 - 4x^3 + 10x^2$. It’s crucial to maintain accuracy in each of these individual multiplications, as errors early in the process can propagate through the rest of the calculation. By carefully applying the distributive property and paying close attention to both coefficients and exponents, we set the stage for the subsequent steps in multiplying the polynomials.

2. Distribute $4x$

Next, we distribute the second term of the first polynomial, $4x$, across the second polynomial $\left(x^2 - 2x + 5\right)$. This involves multiplying $4x$ by each term within the parentheses: $x^2$, $-2x$, and $5$. Similar to the previous step, we apply the power rule and coefficient multiplication to each of these interactions. Multiplying $4x$ by $x^2$ involves multiplying the coefficients (4 * 1 = 4) and adding the exponents of $x$ (1 + 2 = 3), resulting in $4x^3$. When multiplying $4x$ by $-2x$, we multiply the coefficients (4 * -2 = -8) and add the exponents of $x$ (1 + 1 = 2), which gives us $-8x^2$. Finally, multiplying $4x$ by 5 involves multiplying the coefficient 4 by 5, resulting in $20x$. This methodical distribution ensures that we account for each interaction between $4x$ and the terms of the second polynomial. The result of this step is the expression $4x^3 - 8x^2 + 20x$. It’s important to note the alignment of terms with the same exponent as we proceed, as this will simplify the process of combining like terms later on. Careful attention to detail in each multiplication, along with a systematic approach to distribution, ensures accuracy and clarity in the overall process of multiplying polynomials. By consistently applying these principles, we can confidently navigate more complex algebraic expressions.

3. Distribute $-3$

Now, we distribute the third term of the first polynomial, $-3$, across the second polynomial $\left(x^2 - 2x + 5\right)$. This involves multiplying $-3$ by each term within the parentheses: $x^2$, $-2x$, and $5$. This step is slightly simpler than the previous ones, as we are multiplying a constant by each term. When multiplying $-3$ by $x^2$, we simply multiply the coefficient -3 by the implicit coefficient 1 of $x^2$, resulting in $-3x^2$. Multiplying $-3$ by $-2x$ involves multiplying the coefficients (-3 * -2 = 6), giving us $6x$. Finally, multiplying $-3$ by 5 involves multiplying the coefficients (-3 * 5 = -15), resulting in -15. This completes the distribution of the third term, giving us the expression $-3x^2 + 6x - 15$. As with the previous steps, maintaining accuracy and attention to detail is crucial. The careful distribution of each term, including the signs, ensures that we capture all the necessary components for the final simplified expression. By consistently applying this methodical approach, we minimize the risk of errors and build confidence in our ability to manipulate algebraic expressions effectively. With the distribution of $-3$ complete, we are now ready to move on to the final step of combining like terms.

4. Combine Like Terms

After distributing each term, we are left with a series of terms that need to be simplified by combining like terms. Like terms are terms that have the same variable raised to the same power. In our case, the expanded expression from the previous steps is: $2x^4 - 4x^3 + 10x^2 + 4x^3 - 8x^2 + 20x - 3x^2 + 6x - 15$. To combine like terms, we identify terms with the same variable and exponent and then add or subtract their coefficients. For example, $-4x^3$ and $4x^3$ are like terms, and their coefficients sum to zero, effectively canceling each other out. Similarly, $10x^2$, $-8x^2$, and $-3x^2$ are like terms, and their coefficients combine to give $-x^2$. The terms $20x$ and $6x$ are also like terms, combining to $26x$. Finally, the constant term -15 remains unchanged as there are no other constant terms to combine it with. By systematically identifying and combining like terms, we simplify the expression to its most concise form. This process not only reduces the complexity of the expression but also makes it easier to work with in subsequent mathematical operations. The ability to accurately combine like terms is a fundamental skill in algebra, and mastering it is essential for success in more advanced mathematical topics. The final simplified expression represents the product of the original two polynomials and provides a clear and concise representation of their relationship.

Final Result

After meticulously distributing each term and combining like terms, we arrive at the final result of the multiplication: $2x^4 - x^2 + 26x - 15$. This expression represents the product of the original polynomials, $\left(2x^2 + 4x - 3\right)$ and $\left(x^2 - 2x + 5\right)$. The process of arriving at this result involved several key steps, each requiring careful attention to detail. From the initial distribution of terms to the final combination of like terms, accuracy and a systematic approach were paramount. The final expression is a polynomial of degree four, reflecting the highest power of $x$ present in the expression. This result not only provides the answer to the specific multiplication problem but also reinforces the fundamental principles of polynomial multiplication. The ability to confidently perform these operations is crucial for various mathematical applications, including solving equations, simplifying expressions, and even in calculus. Furthermore, understanding the structure and properties of polynomials is essential for advanced mathematical concepts. The final result, $2x^4 - x^2 + 26x - 15$, serves as a testament to the power of systematic algebraic manipulation and provides a solid foundation for future mathematical endeavors. By mastering the techniques involved in polynomial multiplication, we unlock a deeper understanding of algebraic relationships and pave the way for more complex problem-solving.

Conclusion

In conclusion, the multiplication of polynomials, as demonstrated by the example $\left(2x^2 + 4x - 3\right) \left(x^2 - 2x + 5\right)$, is a fundamental skill in algebra. This process requires a systematic application of the distributive property, careful attention to detail, and the ability to combine like terms accurately. The step-by-step approach outlined in this guide provides a clear methodology for tackling such problems, breaking down the seemingly complex task into manageable steps. From distributing each term of one polynomial across the terms of the other to identifying and combining like terms, each stage contributes to the final simplified expression. The result, $2x^4 - x^2 + 26x - 15$, showcases the power of algebraic manipulation and the importance of mastering these foundational skills. Polynomial multiplication is not merely an isolated mathematical operation; it is a building block for more advanced concepts in algebra and calculus. The ability to confidently perform these operations opens doors to a deeper understanding of mathematical relationships and problem-solving techniques. By embracing a systematic approach and practicing diligently, students and enthusiasts alike can develop proficiency in polynomial multiplication and unlock the potential for further mathematical exploration. This skill serves as a valuable asset in various fields, from engineering and physics to computer science and economics, highlighting its relevance beyond the classroom. The journey through polynomial multiplication is a journey of mathematical empowerment, equipping individuals with the tools to navigate complex algebraic expressions with confidence and precision.