Multiplying Polynomials Finding The Product Of (2x² + 5x - 3) And (3x + 7)

Introduction

In the realm of algebra, multiplying polynomials is a fundamental operation. This article delves into the process of finding the product of two polynomials, specifically the quadratic expression $(2x^2 + 5x - 3)$ and the linear expression $(3x + 7)$. We will explore the step-by-step method of polynomial multiplication, providing a clear and concise explanation to enhance your understanding of this essential mathematical concept. This comprehensive guide will not only walk you through the solution but also provide valuable insights into the underlying principles, ensuring you're well-equipped to tackle similar problems with confidence. So, let's embark on this algebraic journey and unravel the intricacies of polynomial multiplication.

Understanding Polynomial Multiplication

At its core, multiplying polynomials involves applying the distributive property repeatedly. The distributive property, a cornerstone of algebra, states that for any numbers a, b, and c: a(b + c) = ab + ac. This principle extends seamlessly to polynomials, where we multiply each term of one polynomial by every term of the other. When multiplying polynomials, it's crucial to maintain organization and meticulous attention to detail. This ensures that all terms are properly accounted for and combined, leading to the accurate determination of the product. By systematically applying the distributive property and combining like terms, we can navigate the complexities of polynomial multiplication and arrive at the correct result. This process is not just a mathematical exercise; it's a skill that builds a strong foundation for advanced algebraic concepts.

Step-by-Step Multiplication Process

To find the product of $(2x^2 + 5x - 3)$ and $(3x + 7)$, we'll meticulously apply the distributive property. This involves multiplying each term of the first polynomial, $(2x^2 + 5x - 3)$, by each term of the second polynomial, $(3x + 7)$. This step-by-step approach ensures that we account for every possible term combination, laying the groundwork for accurate simplification and the final solution.

1. Multiply each term of the first polynomial by $3x$:

   • $$(2x^2 * 3x = 6x^3)$$

   • $$(5x * 3x = 15x^2)$$

   • $$(-3 * 3x = -9x)$$

2. Multiply each term of the first polynomial by $7$:

   • $$(2x^2 * 7 = 14x^2)$$

   • $$(5x * 7 = 35x)$$

   • $$(-3 * 7 = -21)$$

Combining Like Terms

After applying the distributive property, we have a series of terms that need to be simplified. This is where the process of combining like terms comes into play. Like terms are terms that share the same variable raised to the same power. For instance, $15x^2$ and $14x^2$ are like terms because they both contain the variable x raised to the power of 2. Similarly, $-9x$ and $35x$ are like terms. Combining like terms involves adding or subtracting their coefficients while keeping the variable and its exponent unchanged. This step is crucial for condensing the expression into its simplest form, making it easier to interpret and use in further calculations. By carefully identifying and combining like terms, we ensure that the final result is presented in the most concise and understandable manner.

• The result of the multiplication is:

  $$6x^3 + 15x^2 - 9x + 14x^2 + 35x - 21$$

• Combine like terms:

  • 6x^3$ (no like terms)

  • $$15x^2 + 14x^2 = 29x^2$$

  • $$-9x + 35x = 26x$$

  • -21$ (no like terms)

The Final Product

After meticulously applying the distributive property and combining like terms, we arrive at the simplified product of the two polynomials. This final expression represents the result of multiplying $(2x^2 + 5x - 3)$ and $(3x + 7)$, encapsulating the essence of the algebraic operation. The process highlights the importance of precision and attention to detail in mathematical calculations. The final product, a polynomial in its own right, can be used for further algebraic manipulations, such as solving equations, graphing functions, or exploring more complex mathematical relationships. Understanding how to arrive at this final product is a cornerstone of algebraic proficiency.

• Therefore, the product is:

  $$6x^3 + 29x^2 + 26x - 21$$

Conclusion

In conclusion, multiplying polynomials is a fundamental skill in algebra that requires a systematic approach and a solid understanding of the distributive property. Throughout this article, we've meticulously dissected the process of finding the product of $(2x^2 + 5x - 3)$ and $(3x + 7)$, emphasizing the importance of each step. From the initial application of the distributive property to the crucial task of combining like terms, we've highlighted the critical elements that lead to an accurate solution. The final product, $6x^3 + 29x^2 + 26x - 21$, represents the culmination of these steps, showcasing the power of algebraic manipulation. Mastering this process not only enhances your algebraic proficiency but also lays a strong foundation for tackling more complex mathematical challenges. By practicing and internalizing these techniques, you'll be well-equipped to confidently navigate the world of polynomials and beyond.

The correct answer is D. $6x^3 + 29x^2 + 26x - 21$