Problem 134 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the rational expression in simplest form. $$\frac{5 x^{2} y^{2}+25 x^{2} y}{x y+5 x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(5xy\).

Step 1: Factorize the numerator

To simplify a rational function, first factorize the numerator. For \(5x^{2}y^{2} + 25x^{2}y\), factor out the common term to get \(5x^{2}y(y + 5)\).

Step 2: Factorize the denominator

Continue by factorizing the denominator \(xy + 5x\). Factor out the common term 'x' to get \(x(y + 5)\).

Step 3: Simplify the fraction

Now, simplify the expression \(\frac{5x^{2}y(y + 5)}{x(y + 5)}\). The common term ads (y+5) in the numerator and the denominator can be shortened, also divide \(5x^{2}y\) by 'x' to give the final simplified expression.

Key Concepts

Factorization

Factorization is a key step when working with rational expressions. It involves breaking down complex algebraic expressions into products of simpler factors. This process makes it easier to identify common factors in the numerator and the denominator, which can eventually be canceled out. In this particular problem, we start by factorizing both the numerator and the denominator separately.

• For the numerator, \(5x^{2}y^{2} + 25x^{2}y\), we notice that both terms share common factors \(5x^{2}y\). Extracting these common factors results in \(5x^{2}y(y + 5)\).
• Similarly, for the denominator, \(xy + 5x\), the common factor is \(x\), leading us to \(x(y + 5)\).

Factoring at an early stage simplifies the expression and paves the way for the next step in simplification.

Simplification

Simplification is the process of reducing complex expressions to their simplest forms. After the factorization step, simplification becomes straightforward as it primarily involves canceling out common factors. This exercise provides a clear example of how simplification can take a tedious expression and make it much more manageable.
In our expression \(\frac{5x^{2}y(y + 5)}{x(y + 5)}\), both the numerator and the denominator contain the term \((y+5)\). Since they are common factors, they can be canceled from both the top and bottom. Now we are left with the simplified expression \(\frac{5x^{2}y}{x}\).
Canceling the \(x\) gives us \(5xy\). Performing these cancellations leads to the simplest form of the given expression, making it easier to handle.

Numerator and Denominator

A vital component of dealing with rational expressions is understanding the roles of the numerator and the denominator. These are the top and bottom parts of a fraction, respectively.

• The **numerator** is the expression above the fraction line which represents the portion of the whole that is being considered. In our case, the initial numerator is \(5x^{2}y^{2} + 25x^{2}y\).
• The **denominator** is the expression below the fraction line, functioning as the divisor. Here, it begins as \(xy + 5x\).

Both parts need to be considered for factorization to ensure any common terms can be canceled. Simplifying and reducing these expressions into their simplest form often help in solving the problem efficiently and understanding the underlying relationships in the expression.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions (expressions involving variables). Simplifying these fractions involves steps similar to those used with numerical fractions.
In algebraic fractions, we apply factorization to both the numerator and the denominator to seek out common terms. These common terms, once identified, allow us to simplify the fraction by dividing both parts by these terms, similar to how you might with regular numbers.
For example, if we identify \((y+5)\) in both the numerator and the denominator, we can eliminate it from the expression.
This process not only simplifies the expression, making it easier to work with but also reveals deeper mathematical insights into the relationships between the component parts of the fraction.