Problem 24 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{8 x-7}{x+3}+\frac{2 x+37}{x+3}\)

Step-by-Step Solution

Answer

The result is 10.

Step 1: Identify Common Denominator

Both the fractions \( \frac{8x-7}{x+3} \) and \( \frac{2x+37}{x+3} \) have the same denominator \((x+3)\). This makes it straightforward to add them since their denominators are already identical.

Step 2: Add the Numerators

Since the denominators are the same, we can combine the numerators: \((8x - 7) + (2x + 37) \). This will give us a single numerator for our fraction.

Step 3: Simplify the Numerator

Combine like terms in the numerator: \[ \begin{align*} 8x - 7 + 2x + 37 & = (8x + 2x) + (-7 + 37) \& = 10x + 30.\end{align*} \]

Step 4: Write the Simplified Fraction

Place the simplified numerator over the common denominator: \[ \frac{10x + 30}{x+3} \].

Step 5: Factorize and Simplify

Notice that the numerator \( 10x + 30 \) can be factored by pulling out the greatest common factor (GCF) of 10:\[ 10x + 30 = 10(x + 3) \].Thus, the fraction becomes: \[ \frac{10(x+3)}{x+3} \].

Step 6: Cancel the Common Factor

Since \((x+3)\) appears in both the numerator and denominator, they can be canceled, resulting in:\[ 10 \].

Key Concepts

Addition of Fractions

When dealing with addition of fractions, the key factor to consider is whether the denominators are the same. If they are, the process becomes straightforward since you only need to add the numerators and place the result over the common denominator. This approach can simplify calculations, especially when working with variables.

• First, identify if the fractions have a common denominator. If they do, align the fractions for easy addition.
• Add the numerators together while keeping the common denominator intact.

In our exercise, both fractions share the denominator \((x+3)\), therefore, the numerators \(8x-7\) and \(2x+37\) are easily combined. The next step involves simplifying the resulting expression by incorporating techniques like combining like terms.

Factoring Polynomials

Factoring polynomials is a powerful tool for simplifying algebraic expressions and equations. It involves expressing a polynomial as a product of its factors. This can be especially useful when simplifying expressions after operations like addition or subtraction, as it often reveals opportunities for further simplification.

• Look for a common factor in each term of the polynomial. A common factor is a number or variable that divides all terms without a remainder.
• Factor out the greatest common factor (GCF) from the polynomial. The GCF is the largest expression that divides each term of the polynomial completely.

In the provided solution, after adding the numerators to form \(10x + 30\), we notice that both terms share a factor of 10. Factoring this out gives \(10(x + 3)\), greatly simplifying the polynomial and setting the stage for further reduction.

Simplifying Expressions

Simplifying algebraic expressions is about making them as concise and easy to manage as possible. After factoring polynomials and performing operations like addition, you might find opportunities to cancel out terms or simplify the structure of the expression further.

• After factoring, check if any terms or expressions can be canceled. This usually happens when a factor appears in both the numerator and the denominator.
• Remove these common factors to simplify the expression to its simplest form.

In our example, once \(10(x+3)\) is over the common denominator \((x+3)\), the \(x+3\) terms cancel each other out. This leaves us with the simplest possible form: just \(10\). This emphasizes the usefulness of factoring and simplification in achieving the cleanest possible result.