Subtracting Polynomials Lome's Step-by-Step Method

In mathematics, polynomial subtraction is a fundamental operation. Understanding the process is crucial for various algebraic manipulations and problem-solving scenarios. This article delves into the method used by Lome to subtract the polynomial $6x^3 - 2x + 3$ from $-3x^3 + 5x^2 + 4x - 7$. We will break down each step, providing a comprehensive understanding of the underlying principles. This exploration will not only clarify Lome's approach but also equip you with the skills to tackle similar polynomial subtraction problems with confidence. The initial problem Lome faced was to subtract $6x^3 - 2x + 3$ from $-3x^3 + 5x^2 + 4x - 7$. This means we are looking for the result of the expression $(-3x^3 + 5x^2 + 4x - 7) - (6x^3 - 2x + 3)$. The challenge lies in correctly handling the subtraction of terms, particularly the signs. Polynomial subtraction involves distributing the negative sign across the terms of the polynomial being subtracted and then combining like terms. This requires careful attention to detail to avoid errors. The goal is to simplify the expression into a standard polynomial form, where terms are arranged in descending order of their exponents. This article aims to provide a clear, step-by-step explanation of this process, ensuring that you can confidently perform polynomial subtraction.

Step 1 Transforming Subtraction into Addition

Lome's initial step involves transforming the subtraction problem into an addition problem. This is achieved by distributing the negative sign across the second polynomial. The original problem is to subtract $(6x^3 - 2x + 3)$ from $(-3x^3 + 5x^2 + 4x - 7)$. To transform this into an addition problem, we rewrite the expression as $(-3x^3 + 5x^2 + 4x - 7) + (-1)(6x^3 - 2x + 3)$. This transformation is crucial because it allows us to apply the rules of addition, which are often more straightforward to handle. The next substep is to distribute the $-1$ across the terms inside the second parenthesis. This means multiplying each term in $(6x^3 - 2x + 3)$ by $-1$. When we multiply $6x^3$ by $-1$, we get $-6x^3$. When we multiply $-2x$ by $-1$, we get $+2x$. And when we multiply $+3$ by $-1$, we get $-3$. Therefore, $(-1)(6x^3 - 2x + 3)$ becomes $-6x^3 + 2x - 3$. Now, we can rewrite the entire expression as $(-3x^3 + 5x^2 + 4x - 7) + (-6x^3 + 2x - 3)$. This form is more manageable because it eliminates the subtraction operation, replacing it with addition, which simplifies the subsequent steps. This transformation is a fundamental technique in polynomial arithmetic, making it easier to combine like terms and simplify the expression. By converting subtraction to addition, Lome sets the stage for the next steps, which involve rearranging and combining like terms.

Step 2 Rearranging Terms for Clarity

The next critical step Lome takes is to rearrange the terms to group like terms together. This rearrangement makes it easier to identify and combine terms with the same variable and exponent. From the previous step, we have the expression $(-3x^3 + 5x^2 + 4x - 7) + (-6x^3 + 2x - 3)$. To rearrange the terms, we remove the parentheses and group the terms with the same variable and exponent. This involves writing out all the terms and then reorganizing them based on their degree. The $x^3$ terms are $-3x^3$ and $-6x^3$. The $x^2$ term is $5x^2$. The $x$ terms are $4x$ and $2x$. The constant terms are $-7$ and $-3$. By grouping these terms, we get $-3x^3 - 6x^3 + 5x^2 + 4x + 2x - 7 - 3$. This rearrangement is a crucial organizational step. It helps in visualizing which terms can be combined and prevents errors that might occur if terms are combined incorrectly. The associative and commutative properties of addition allow us to rearrange the terms without changing the value of the expression. This step is particularly important when dealing with polynomials that have many terms, as it simplifies the process of combining like terms. This systematic approach reduces the cognitive load and makes the process less error-prone. By carefully rearranging the terms, Lome sets the stage for the final simplification step, which involves combining the like terms to obtain the final result. The organized arrangement allows for a clear and efficient calculation of the polynomial's simplified form.

Step 3 Combining Like Terms for Simplification

After rearranging the terms, the final step is to combine the like terms. This involves adding the coefficients of terms with the same variable and exponent. From the rearranged expression $-3x^3 - 6x^3 + 5x^2 + 4x + 2x - 7 - 3$, we can now combine the terms. First, we combine the $x^3$ terms: $-3x^3 - 6x^3$. Adding the coefficients $-3$ and $-6$, we get $-9$. So, the combined term is $-9x^3$. Next, we have the $x^2$ term, which is $5x^2$. Since there are no other $x^2$ terms, it remains as $5x^2$. Then, we combine the $x$ terms: $4x + 2x$. Adding the coefficients $4$ and $2$, we get $6$. So, the combined term is $6x$. Finally, we combine the constant terms: $-7 - 3$. Adding these, we get $-10$. Putting these combined terms together, the simplified polynomial is $-9x^3 + 5x^2 + 6x - 10$. This step is crucial for simplifying the polynomial into its most concise form. Combining like terms reduces the complexity of the expression, making it easier to understand and use in further calculations. It is a fundamental operation in algebra and is used extensively in various mathematical contexts. This process ensures that the polynomial is expressed in its simplest form, with each term representing a unique degree of the variable. By carefully combining like terms, Lome arrives at the final simplified polynomial, which represents the difference between the two original polynomials. This final result is a clear and concise representation of the subtraction, making it easier to interpret and use in subsequent mathematical operations.

Final Answer

Therefore, after carefully following the steps of transforming subtraction into addition, rearranging terms for clarity, and combining like terms for simplification, the final result of subtracting $6x^3 - 2x + 3$ from $-3x^3 + 5x^2 + 4x - 7$ is $-9x^3 + 5x^2 + 6x - 10$. This process highlights the importance of methodical steps in polynomial arithmetic. Each step, from distributing the negative sign to combining like terms, plays a crucial role in achieving the correct result. Understanding and practicing these steps will enhance your ability to handle more complex algebraic problems. The journey from the initial subtraction problem to the final simplified polynomial underscores the elegance and precision of algebraic manipulation. By breaking down the problem into manageable steps, Lome's method provides a clear and effective way to perform polynomial subtraction. This understanding is not only beneficial for academic purposes but also for various real-world applications where algebraic expressions need to be simplified. The ability to confidently subtract polynomials is a valuable skill in mathematics and related fields.