Problem 35 - Elementary and Intermediate Algebra

Question

Divide the polynomial by the monomial. See Example 2. $$ \frac{-25 x^{2} y^{3}+30 x y^{2}-5 x y}{-5 x^{2} y^{2}} $$

Step-by-Step Solution

Answer

The answer is \(5y - \frac{6}{x} + \frac{1}{xy}\).

Step 1: Simplify Each Term

Divide each term of the polynomial by the monomial \(-5x^2y^2\).1. **First term: **\(-25x^2y^3 \div -5x^2y^2 \)2. **Second term: **\(30xy^2 \div -5x^2y^2 \)3. **Third term: **\(-5xy \div -5x^2y^2\)

Step 2: Calculate the First Term

For the first term, simplify:- Coefficients: \(-25 \div -5 = 5\)- Variables: \(x^2 \div x^2 = 1\) and \(y^3 \div y^2 = y\)So, \(-25x^2y^3 \div -5x^2y^2 = 5y\).

Step 3: Calculate the Second Term

For the second term, simplify:- Coefficients: \(30 \div -5 = -6\)- Variables: \(x \div x^2 = \frac{1}{x}\) and \(y^2 \div y^2 = 1\)So, \(30xy^2 \div -5x^2y^2 = -\frac{6}{x}\).

Step 4: Calculate the Third Term

For the third term, simplify:- Coefficients: \(-5 \div -5 = 1\)- Variables: \(x \div x^2 = \frac{1}{x}\) and \(y \div y^2 = \frac{1}{y}\)So, \(-5xy \div -5x^2y^2 = \frac{1}{xy}\).

Step 5: Form the Final Answer

Combine the simplified terms to form the final expression: \(5y - \frac{6}{x} + \frac{1}{xy}\).

Key Concepts

Understanding Monomials

In algebra, a monomial is a simple algebraic expression consisting of one term. This term includes three main parts: a coefficient, variables, and their exponents. A monomial can be a single number, a single variable, or a product of numbers and variables. For example, in the expression \(5x^2y^3\), 5 is the coefficient, \(x^2y^3\) includes the variables \(x\) and \(y\) with their respective exponents.

Monomials are immensely useful when performing operations like multiplication or division within polynomials, which are expressions made up of many monomials.

• ### Components of a Monomial
  • **Coefficient:** Represents the number in front of the variables. It tells us how many times to multiply the product of the variables.
  • **Variable(s):** A symbol like \(x\) or \(y\) that represents an unknown quantity.
  • **Exponent(s):** Shows how many times the variable is used in the multiplication. For example, \(x^2\) means \(x\) multiplied by itself.

When dividing by a monomial, you have to pay attention to each part of it: the coefficient, variables, and exponents to ensure proper division of each part within the polynomial. This helps in reaching the correct simplified expression.

Simplifying Expressions

Simplifying algebraic expressions makes them easier to work with. In polynomials, which consist of several monomials, simplification involves performing operations like addition, subtraction, multiplication, and division. The aim is to reduce the expression to its simplest form.

Here's how simplification often works:

• **Dividing Coefficients:** Start by dividing the numerical coefficients in each term. If we have \-25 \div \-5, the result is 5 because dividing two negative numbers yields a positive result.


• **Reducing Variables:** Divide powers of the same base by subtracting the exponents. For example, \(x^2 \div x^2 = x^{2-2} = x^0 = 1\). This principle also applies to other variables like \(y\).


• **Combining Simplified Terms:** Once you've simplified each term, you combine them to form the polynomial's simplified version. This makes it easier to interpret and further manipulate.

With division or other operations, ensuring each step observes mathematical rules guarantees the accuracy of the final outcome.

The Role of Variables in Algebra

Variables play a critical role in algebra. They are symbols, typically letters like \(x\), \(y\), and \(z\), used to represent unknown or changing values. In polynomial division, these variables carry specific rules when undergoing operations.

• **Symbolic Representation:** Variables help in formulating general equations and expressions that can be solved or manipulated to find various values or solve problems.
• **Operations with Variables:** Just like numbers, variables can be added, subtracted, multiplied, and divided, provided you follow algebraic rules. For example, dividing \(x\) by \(x^2\) results in \(rac{1}{x}\).

Variables, when paired with coefficients and exponents, form the structure of polynomials and monomials. Understanding how to handle them, especially when dividing, ensures you can simplify equations effectively and solve versatile algebraic problems.

In essence, grasping the use of variables unlocks the ability to perform complex algebraic manipulations, enhancing your capacity to approach diverse mathematical challenges.