Multiplying Polynomials A Step-by-Step Guide To (x² + 3)(x³ - 5)

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Polynomial multiplication is a fundamental concept in algebra, forming the bedrock for more advanced mathematical operations. In this comprehensive guide, we will delve into the step-by-step process of multiplying two polynomials, specifically (x² + 3) and (x³ - 5). Our goal is not just to arrive at the correct answer but to provide a clear, concise explanation that empowers you to tackle similar problems with confidence.

Understanding Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial across all terms of the other polynomial. This process relies on the distributive property of multiplication, which states that a(b + c) = ab + ac. In essence, we're ensuring that every term in the first polynomial interacts with every term in the second polynomial.

When multiplying polynomials, it's crucial to pay close attention to the signs of the terms. A positive term multiplied by a positive term yields a positive term, while a positive term multiplied by a negative term (or vice versa) results in a negative term. Moreover, when multiplying variables with exponents, we add the exponents together. For example, x² multiplied by x³ gives us x^(2+3) = x⁵. This rule stems from the fundamental principle of exponents, which dictates how powers of the same base interact during multiplication.

The process of polynomial multiplication is akin to a systematic distribution, ensuring that each term plays its role in shaping the final product. By meticulously applying the distributive property and the rules of exponents, we can navigate the intricacies of polynomial multiplication with precision and accuracy. This foundational understanding paves the way for more complex algebraic manipulations and problem-solving endeavors.

Step 1: Distribute the First Term

Begin by distributing the first term of the first polynomial (x²) across both terms of the second polynomial (x³ - 5). This means multiplying x² by both x³ and -5:

x² * x³ = x⁵ x² * -5 = -5x²

In this initial step, we're essentially laying the groundwork for the subsequent calculations. The distribution of x² across the terms of the second polynomial yields two distinct terms: x⁵ and -5x². These terms represent the partial products resulting from the interaction of x² with x³ and -5, respectively. The meticulous execution of this step is paramount, as it sets the stage for the accurate computation of the final product. The distributive property serves as our guiding principle, ensuring that each term in the first polynomial is systematically multiplied by each term in the second polynomial, thereby capturing all possible interactions and laying the foundation for a comprehensive solution.

Step 2: Distribute the Second Term

Next, distribute the second term of the first polynomial (3) across both terms of the second polynomial (x³ - 5):

3 * x³ = 3x³ 3 * -5 = -15

Continuing our systematic approach, we now shift our focus to the second term of the first polynomial, which is 3. We repeat the distribution process, this time multiplying 3 by both x³ and -5. This yields two additional terms: 3x³ and -15. These terms represent the partial products resulting from the interaction of 3 with x³ and -5, respectively. Just as in the previous step, the meticulous execution of this step is crucial for ensuring the accuracy of the final result. By systematically distributing each term of the first polynomial across the terms of the second polynomial, we ensure that all possible combinations are accounted for, laying the groundwork for a complete and accurate solution.

Step 3: Combine Like Terms

Now, combine the terms obtained in the previous steps:

x⁵ - 5x² + 3x³ - 15

Upon completing the distribution process, we arrive at a collection of terms that represent the partial products of our polynomial multiplication. However, our task is not yet complete. The next crucial step involves combining like terms, which are terms that share the same variable and exponent. In our case, we have x⁵, -5x², 3x³, and -15. Upon closer inspection, we observe that there are no other terms with the same variable and exponent as x⁵, -5x², 3x³, and -15. This means that there are no like terms to combine in this particular instance. However, it's essential to remember that combining like terms is a fundamental step in simplifying polynomial expressions. By identifying and combining terms with the same variable and exponent, we can reduce the complexity of the expression and arrive at a more concise and manageable form. In scenarios where like terms are present, their combination is paramount for achieving the most simplified representation of the polynomial expression.

Step 4: Write in Standard Form

It's customary to write polynomials in standard form, where the terms are arranged in descending order of their exponents:

x⁵ + 3x³ - 5x² - 15

While the expression x⁵ - 5x² + 3x³ - 15 accurately represents the product of the polynomials, it is not in the conventional standard form. In standard form, polynomials are arranged in descending order of their exponents, which means that the term with the highest exponent is written first, followed by the term with the next highest exponent, and so on. This convention enhances the readability and comparability of polynomials, facilitating easier analysis and manipulation. To adhere to standard form, we rearrange the terms of our expression as follows: x⁵ + 3x³ - 5x² - 15. This arrangement presents the polynomial in a clear and organized manner, aligning with the established mathematical convention and facilitating further operations or analyses that may be required. Adhering to standard form is not merely an aesthetic preference; it is a fundamental practice that promotes clarity, consistency, and ease of communication in mathematical expressions.

The Correct Answer

Therefore, the correct product of (x² + 3)(x³ - 5) is:

B. x⁵ + 3x³ - 5x² - 15

Why Other Options Are Incorrect

Let's briefly examine why the other options are incorrect:

  • A. X⁵ - 15: This option only considers the product of the first terms (x² and x³) and the last terms (3 and -5), neglecting the cross-multiplication terms.
  • C. x⁶ - 15: This option incorrectly multiplies the exponents when multiplying x² and x³ (2 + 3 = 5, not 6) and also omits the cross-multiplication terms.
  • D. x⁶ + 3x³ - 5x² - 15: Similar to option C, this option incorrectly multiplies the exponents of x² and x³.

Understanding why incorrect options are wrong is as crucial as knowing the correct answer. By dissecting the errors in each option, we reinforce our understanding of the correct procedure and solidify our grasp of the underlying mathematical principles. Option A, for instance, falls short by neglecting the crucial cross-multiplication terms, which arise from the distribution of each term in one polynomial across all terms in the other. Option C commits an error in exponent arithmetic, incorrectly multiplying the exponents instead of adding them. Similarly, option D mirrors this exponent error, further emphasizing the importance of adhering to the rules of exponent manipulation. By scrutinizing these common pitfalls, we sharpen our analytical skills and cultivate a deeper appreciation for the intricacies of polynomial multiplication. This holistic approach not only equips us to solve the current problem but also fortifies our ability to navigate future mathematical challenges with greater confidence and accuracy.

Conclusion

Mastering polynomial multiplication is essential for success in algebra and beyond. By following these steps and understanding the underlying principles, you can confidently tackle similar problems and build a strong foundation in mathematics. Remember, practice makes perfect, so don't hesitate to work through additional examples to solidify your understanding. Embrace the challenge, and you'll find that polynomial multiplication becomes second nature in no time.