Problem 47 - Algebra 1: Concepts and Skills

Question

Simplify \(\frac{x^{2}}{x+5}-\frac{25}{x+5}\). F. \(\frac{1}{x-5}\) G. \(\frac{x^{2}-25}{x+5}\) H. \(x-5\) J. \(\frac{x-5}{x+5}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{x^{2}}{x+5}-\frac{25}{x+5}\) is \(x - 5\). So, the answer is H. \(x - 5\).

Step 1: Combine The Fractions

Because the fractions \(\frac{x^{2}}{x+5}\) and \(\frac{25}{x+5}\) have the same denominators, they can be combined into one fraction: \(\frac{x^{2}-25}{x+5}\).

Step 2: Simplify The Numerator By Applying the Difference of Squares Formula

In the numerator, \(x^{2}-25\) is a difference of squares and such expressions can be factored with the formula \(a^{2}-b^{2}=(a - b)(a + b)\). Therefore, the numerator \(x^{2}-25\) can be factored as \((x - 5)(x + 5)\). Applying this, the fraction becomes: \(\frac{(x-5)(x+5)}{x+5}\).

Step 3: Cancel Simplifiable Terms

(x + 5) is in both the numerator and the denominator, therefore they cancel out since any number divided by itself equals 1. After canceling, the simplified result is: \(x - 5\).

Key Concepts

Difference of Squares

In algebra, a difference of squares is a special type of expression that can be simplified easily. It’s presented as two terms squared, separated by a minus sign. For example, an expression like \(a^{2} - b^{2}\) is a difference of squares. The beauty of these expressions is they can be factored using the formula:

• \(a^{2} - b^{2} = (a - b)(a + b)\)

This formula tells us that any difference of squares can be broken down into two factors: one with subtraction and the other with addition.

In the original exercise, the numerator \(x^{2} - 25\) is a difference of squares, where \(x^{2}\) is \((x)^{2}\) and \(25\) is \((5)^{2}\). When using the formula, it simplifies to \((x-5)(x+5)\), making the simplification process straightforward and efficient.

Factoring

Factoring is a crucial technique in algebra. It involves writing an expression as a product of its factors. These are quantities that divide evenly into the expression, much like how 2 can be factored from 6 since \(2 \times 3 = 6\).

In the context of the problem at hand, once we recognize the difference of squares in the numerator \(x^{2} - 25\), we can factor it. We already learned it factors to \((x-5)(x+5)\). This breaks down the polynomial into simpler terms, highlighting the portions that can be further simplified or canceled out in the rational expression.

• This process makes complex expressions easier to work with, allows for the cancellation of like terms, and simplifies both mathematical solutions and real-world problems.

Factoring is thus not just a skill but a strategy for making algebra more manageable.

Combining Fractions

Combining fractions is a method where you merge two fractions into one. If the two fractions have the same denominator, the process is as simple as adding or subtracting the numerators.

Take our exercise as an example:

• We have the fractions \(\frac{x^{2}}{x+5}\) and \(\frac{25}{x+5}\).
• Both have the common denominator of \(x+5\).
• Since they share this denominator, we can directly subtract the numerators: \(x^{2} - 25\).

Once combined, this unified fraction can be further simplified by factoring and canceling any common terms. This simplification in turn makes complex expressions easier to solve, and it also facilitates a clearer understanding of the relationships between different mathematical terms and operations.