Polynomial Division Explained Classifying Expressions After Division

When dealing with polynomial division, it's crucial to understand the different types of expressions that can result. In this article, we will dive deep into the process of dividing the quadratic expression (4x² + 15x + 40) by the linear expression (x + 5) using long division. Our primary goal is to identify the correct classification of the resulting expression: Is it a trinomial, a monomial, a polynomial, or something else entirely? This exploration will not only provide the answer but also enhance your understanding of polynomial division and expression classification.

Performing Long Division: A Step-by-Step Approach

To accurately determine the nature of the resulting expression, we must first execute the long division process. Long division, in the context of polynomials, is a method used to divide one polynomial by another polynomial of lower or equal degree. This process mirrors the long division method used with numbers, but instead of digits, we are working with algebraic terms. Let’s break down the steps involved in dividing (4x² + 15x + 40) by (x + 5).

Step 1: Setting Up the Division

The first step involves setting up the long division problem. The dividend, which is (4x² + 15x + 40), goes inside the division symbol, and the divisor, (x + 5), goes outside. This setup visually organizes the problem, making it easier to follow the subsequent steps. It’s akin to setting up a traditional long division problem with numbers, ensuring that each term is correctly aligned for the division process. Proper setup is crucial as it minimizes errors and allows for a clear, step-by-step solution.

Step 2: Dividing the Leading Terms

The core of polynomial long division lies in dividing the leading terms. We start by dividing the leading term of the dividend, which is 4x², by the leading term of the divisor, which is x. When 4x² is divided by x, the result is 4x. This 4x becomes the first term of the quotient, the expression that represents the result of the division. This step is pivotal as it initiates the process of breaking down the dividend into manageable parts, gradually revealing the quotient. The focus on leading terms simplifies the process, making it easier to handle complex polynomial expressions.

Step 3: Multiplying the Quotient Term by the Divisor

Once we have the first term of the quotient, 4x, we multiply it by the entire divisor, (x + 5). This multiplication yields 4x * (x + 5) = 4x² + 20x. This step is essential because it allows us to subtract a portion of the dividend that is divisible by the divisor. By multiplying the quotient term with the divisor, we effectively determine how much of the dividend can be accounted for by the divisor at this stage. The result, 4x² + 20x, is then strategically placed beneath the dividend, aligning like terms to facilitate the next step of subtraction.

Step 4: Subtracting and Bringing Down the Next Term

The next critical step involves subtracting the result obtained in Step 3, which is (4x² + 20x), from the corresponding terms in the dividend, (4x² + 15x). Performing this subtraction, we get (4x² + 15x) - (4x² + 20x) = -5x. Following the subtraction, we bring down the next term from the dividend, which is +40. This process mirrors traditional long division, where remainders are carried down to continue the division. Bringing down the next term ensures that we consider all parts of the dividend in the division process, maintaining the integrity of the solution. The new expression to work with is now -5x + 40, which sets the stage for the next iteration of the division.

Step 5: Repeating the Division Process

We now repeat the process, focusing on the new leading term, -5x. We divide -5x by the leading term of the divisor, x, which gives us -5. This -5 becomes the next term in the quotient. The iterative nature of this process is fundamental to long division, allowing us to systematically break down the dividend until we reach a remainder or a point where further division is not possible. By consistently applying the same steps—dividing, multiplying, subtracting, and bringing down—we can accurately determine the quotient and remainder of any polynomial division problem.

Step 6: Final Multiplication and Subtraction

Following the pattern, we multiply the new quotient term, -5, by the divisor, (x + 5). This yields -5 * (x + 5) = -5x - 25. We then subtract this result from the current expression, (-5x + 40). The subtraction is performed as follows: (-5x + 40) - (-5x - 25) = 65. This step is crucial for determining the final remainder and completing the long division process. The careful subtraction ensures that we account for all terms and signs, leading to an accurate remainder. The result of this subtraction, 65, represents the remainder of the division, which will be used in the final expression of the result.

Step 7: Expressing the Result

After completing the long division, we express the result as the quotient plus the remainder divided by the divisor. In this case, the quotient is 4x - 5, and the remainder is 65. Therefore, the final expression is 4x - 5 + 65/(x + 5). This final expression encapsulates the complete result of the polynomial division, clearly showing both the quotient and the remainder. The quotient represents the whole part of the division, while the remainder term indicates the portion that could not be evenly divided. This comprehensive representation is essential for understanding the full scope of the division's outcome.

Classifying the Resulting Expression

Now that we have the result of the division, which is 4x - 5 + 65/(x + 5), we can classify it based on its terms and structure. The critical part of this expression is the term 65/(x + 5). This term indicates that the expression is not a polynomial due to the presence of a variable in the denominator. Let’s delve deeper into the definitions of different types of expressions to understand this classification better.

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, a polynomial does not have variables in the denominator or under a radical. The expression 4x - 5 is a polynomial because it consists of terms with non-negative integer exponents and no variables in the denominator.

Monomials, Binomials, and Trinomials

Polynomials can be further classified based on the number of terms they contain:

• A monomial is a polynomial with one term (e.g., 3x²).
• A binomial is a polynomial with two terms (e.g., 2x + 1).
• A trinomial is a polynomial with three terms (e.g., x² - 3x + 2).

The expression 4x - 5 is a binomial because it has two terms. However, the entire result, 4x - 5 + 65/(x + 5), is not a polynomial because of the 65/(x + 5) term.

Why the Result is Not a Polynomial

The presence of the term 65/(x + 5) is what disqualifies the entire expression from being a polynomial. This term represents a rational expression where the variable x is in the denominator. Polynomials, by definition, cannot have variables in the denominator. This distinction is crucial in algebraic classification, as it determines the properties and operations that can be applied to the expression. Therefore, while 4x - 5 is a polynomial, the complete result of the division is not.

Conclusion: The Correct Classification

Based on our long division and subsequent analysis, the statement that BEST describes the answer when (4x² + 15x + 40) is divided by (x + 5) is:

The answer is not a polynomial.

This conclusion is drawn from the presence of the term 65/(x + 5) in the result, which violates the definition of a polynomial. Understanding the nuances of polynomial division and expression classification is essential for success in algebra and beyond. By meticulously performing the long division and carefully analyzing the resulting expression, we have confidently determined its correct classification.

By understanding these concepts, you can confidently tackle similar problems and further enhance your algebraic skills. Polynomial division is a foundational topic in algebra, and mastering it will open doors to more advanced mathematical concepts.