Identifying Polynomial Division Problems A Comprehensive Guide

Polynomial division is a fundamental concept in algebra, crucial for simplifying complex expressions and solving equations. Before diving into specific problems, it's essential to understand what constitutes a polynomial and the rules governing its division. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. When we talk about polynomial division, we are essentially looking for cases where the division results in another polynomial, meaning the quotient has only non-negative integer exponents. This often involves distributing the denominator across multiple terms in the numerator or using long division techniques. However, not all division problems yield polynomial results. The presence of variables in the denominator can often lead to non-polynomial expressions, especially when these variables cannot be canceled out through simplification. Thus, identifying which division problem will result in a polynomial requires a careful examination of the given expressions and an understanding of polynomial properties. In the context of the provided examples, we will explore each division problem to determine if the outcome is a polynomial expression or not. This involves simplifying the expressions and checking if any terms have negative or fractional exponents, which would disqualify them from being polynomials. Recognizing patterns and applying the rules of exponents are key skills in determining the nature of the result.

Problem 1: $\frac{6x^2 - 5y + 7}{x + 9}$

The first division problem presents us with the expression $\frac{6x^2 - 5y + 7}{x + 9}$. To determine if this division results in a polynomial, we need to examine the structure of both the numerator and the denominator. The numerator, $6x^2 - 5y + 7$, is a polynomial expression with terms involving $x^2$, $y$, and a constant term. The denominator, $x + 9$, is also a polynomial, a simple binomial. However, the crucial question is whether dividing the numerator by the denominator will yield another polynomial. In this case, we cannot directly simplify the expression by canceling out common factors. The presence of different variables ($x$ and $y$) and the lack of an obvious factorization method suggest that long division would be necessary to simplify this expression. When performing polynomial long division, the result will be a quotient and potentially a remainder. If the remainder is non-zero, the result will include a fractional term with $x + 9$ in the denominator, making the entire expression non-polynomial. Since the numerator does not have a factor of $(x + 9)$, we can anticipate that the division will not result in a clean polynomial. The term $-5y$ in the numerator further complicates matters, as it cannot be easily combined with the terms involving $x$ in the denominator. Therefore, without performing the full long division, we can infer that this division problem is unlikely to result in a polynomial due to the incompatibility of the terms and the structure of the expression. In summary, the initial assessment indicates that $\frac{6x^2 - 5y + 7}{x + 9}$ will not yield a polynomial result.

Problem 2: $(50x^2y + 25xy^2 + 75xy) \div (5xy)$

The second division problem involves dividing the expression $(50x^2y + 25xy^2 + 75xy)$ by $(5xy)$. This problem differs significantly from the first one because the denominator is a single term, which allows us to distribute the division across each term in the numerator. To determine if this division results in a polynomial, we can rewrite the expression as a sum of fractions: $\frac{50x^2y}{5xy} + \frac{25xy^2}{5xy} + \frac{75xy}{5xy}$. Now, we can simplify each fraction individually by canceling out common factors. For the first term, $\frac{50x^2y}{5xy}$, we can divide 50 by 5 to get 10, $x^2$ by $x$ to get $x$, and $y$ by $y$ to get 1. Thus, the simplified term is $10x$. For the second term, $\frac{25xy^2}{5xy}$, we divide 25 by 5 to get 5, $x$ by $x$ to get 1, and $y^2$ by $y$ to get $y$. This simplifies to $5y$. Finally, for the third term, $\frac{75xy}{5xy}$, we divide 75 by 5 to get 15, and both $x$ and $y$ cancel out, leaving us with 15. Combining these simplified terms, we get $10x + 5y + 15$. This expression is a polynomial because all the exponents of the variables are non-negative integers. The simplified expression contains terms with $x$ and $y$ to the power of 1, as well as a constant term. Therefore, the division of $(50x^2y + 25xy^2 + 75xy)$ by $(5xy)$ results in a polynomial. This outcome highlights the importance of recognizing when division can be simplified term by term, leading to a polynomial result.

Problem 3: $\frac{2x^2 - 12x + 43}{x - 3}$

The third division problem asks us to analyze the expression $\frac{2x^2 - 12x + 43}{x - 3}$ and determine if it results in a polynomial. Similar to the first problem, this involves dividing a polynomial by another polynomial. The numerator, $2x^2 - 12x + 43$, is a quadratic polynomial, and the denominator, $x - 3$, is a linear polynomial. To ascertain whether this division results in a polynomial, we need to consider the possibility of simplification through factoring or polynomial long division. First, let's examine if the numerator can be factored in a way that includes the denominator as a factor. If the numerator had a factor of $(x - 3)$, we could cancel it out, resulting in a polynomial. However, upon inspection, the numerator does not readily factor in this way. The quadratic formula or completing the square could be used to find the roots of the numerator, but they are unlikely to be simple integers or fractions that would allow for easy factorization with $(x - 3)$. Given the difficulty in factoring, the next approach is to consider polynomial long division. Performing long division will give us a quotient and a remainder. If the remainder is zero, the result is a polynomial. If the remainder is non-zero, the result will include a fractional term with $(x - 3)$ in the denominator, making the expression non-polynomial. When we perform the long division of $2x^2 - 12x + 43$ by $x - 3$, we obtain a quotient and a non-zero remainder. This indicates that the division does not result in a polynomial. The presence of a remainder term with $x - 3$ in the denominator means the expression is not a polynomial. Therefore, the division problem $\frac{2x^2 - 12x + 43}{x - 3}$ does not yield a polynomial.

Problem 4: $(80x^2y^2 - 32xy^2 - 64y^2) \div (8xy)$

The fourth division problem involves the expression $(80x^2y^2 - 32xy^2 - 64y^2)$ divided by $(8xy)$. Similar to the second problem, this division can be approached by distributing the denominator across each term in the numerator. This method allows us to simplify each term individually and then combine the results. To determine if this division results in a polynomial, we can rewrite the expression as a sum of fractions: $\frac{80x^2y^2}{8xy} - \frac{32xy^2}{8xy} - \frac{64y^2}{8xy}$. Now, let's simplify each fraction. For the first term, $\frac{80x^2y^2}{8xy}$, we divide 80 by 8 to get 10, $x^2$ by $x$ to get $x$, and $y^2$ by $y$ to get $y$. Thus, the simplified term is $10xy$. For the second term, $\frac{32xy^2}{8xy}$, we divide 32 by 8 to get 4, $x$ by $x$ to get 1, and $y^2$ by $y$ to get $y$. This simplifies to $4y$. For the third term, $\frac{64y^2}{8xy}$, we divide 64 by 8 to get 8, $y^2$ by $y$ to get $y$, but we are left with an $x$ in the denominator, resulting in $\frac{8y}{x}$. Combining these simplified terms, we have $10xy - 4y - \frac{8y}{x}$. This expression is not a polynomial because the last term, $\frac{8y}{x}$, has a variable in the denominator. The presence of $x$ in the denominator means that the exponent of $x$ is effectively -1, which violates the condition for a polynomial having only non-negative integer exponents. Therefore, the division of $(80x^2y^2 - 32xy^2 - 64y^2)$ by $(8xy)$ does not result in a polynomial due to the presence of a term with a variable in the denominator. This problem illustrates that while distributing the division can simplify terms, it is crucial to check for any resulting terms that might not conform to the definition of a polynomial.

In conclusion, after analyzing the four division problems, only the second problem, $(50x^2y + 25xy^2 + 75xy) \div (5xy)$, results in a polynomial. This is because the division can be simplified term by term, and all resulting terms have non-negative integer exponents. The other problems either require long division, which leads to a non-zero remainder and a fractional term, or result in terms with variables in the denominator, thus violating the conditions for a polynomial. Understanding the properties of polynomials and the rules of division is crucial for solving these types of problems effectively. Identifying whether a division problem will result in a polynomial involves careful examination of the expressions and the application of simplification techniques.