Problem 14 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{x^{2}+5 x}{x y}$$

Step-by-Step Solution

Answer

\(\frac{x+5}{y}\)

Step 1: Identify Common Factors

Look at the numerator of the fraction \(x^2 + 5x\). Notice that both terms in the numerator, \(x^2\) and \(5x\), have \(x\) as a common factor.

Step 2: Factor Out the Common Factor

Factor \(x\) out of the numerator \(x^2 + 5x\). This gives:\[x(x + 5)\]

Step 3: Simplify the Fraction

Substitute the factored numerator back into the fraction in place of \(x^2 + 5x\):\[\frac{x(x + 5)}{xy}\]Now, cancel the common \(x\) from the numerator and the denominator:\[\frac{x+5}{y}\]Make sure \(x eq 0\) and \(y eq 0\) to avoid any division by zero.

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the pieces of a puzzle within a mathematical expression. It involves breaking down a polynomial into smaller, more manageable factors that, when multiplied together, give you the original expression. In our exercise, the polynomial is the numerator of the fraction, which is \(x^2 + 5x\). Here are the steps and tricks you can use for factoring:

• Identify Common Factors: This is your first step. In any polynomial, check if there is a common factor shared by the terms. In our example, both terms contain the variable \(x\).
• Factor Out: Once you spot a common factor, you "factor it out." This means you'll divide each term by this factor, effectively pulling it out in front of a set of parentheses. The factored form of \(x^2 + 5x\) becomes \(x(x + 5)\).
• Verification: Always multiply back the factors to ensure they return to the original expression. This step validates the accuracy of your factorization.

Understanding how to factor polynomials is an essential skill in simplifying expressions and solving equations.

Simplifying Expressions

Simplifying expressions is all about making them easier to read and interpret by reducing them to their simplest form. The process can involve multiple techniques including factoring, reducing fractions, and cancelling common terms.

For algebraic fractions, follow these steps:

• Factor the parts: As we factored the numerator, we opened the door for simplification.
• Cancel Common Terms: Look for common terms that appear in both the numerator and denominator. In the simplified fraction \(\frac{x(x + 5)}{xy}\), the \(x\) cancels out, leaving \(\frac{x + 5}{y}\).
• Simplify further: If possible, continue this process until the expression can’t be reduced any further.

Take care to only cancel terms that are factors of the whole numerator or denominator, rather than parts within a parenthesis or additions/subtractions.

Division by Zero

Division by zero is something to watch out for in expressions and equations. It is important because division by zero is undefined in mathematics. This means it cannot be performed or lead to a meaningful result. Always consider this when simplifying.Here's why and how to handle it:

• Identifying Zero Division Risk: In fractions, check the denominator of the original and simplified expressions for any potential zero values.
• Preventing Division by Zero: Declare restrictions such as \(x eq 0\) and \(y eq 0\) based on your expression. This tells us that these variables cannot be zero without causing a mathematical issue.
• Understanding Implications: If a variable in a denominator equals zero in a real-world context, it might indicate an incomplete model or overlooked constraint.

By watching out for zero in your denominators, you can avoid undefined expressions and better ensure your solutions are both meaningful and accurate.