Problem 28 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$ \frac{x^{2}+25}{2 x+10} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{x^{2}+25}{2(x + 5)}\).

Step 1: Identifying common factors in numerator and denominator

The numerator of the fraction is \(x^{2}+25\), and the denominator is \(2x+10\). Both terms in the denominator are divisible by 2, so we can factor out a 2.

Step 2: Factorizing the expression

After factoring out 2 from the denominator, the expression now resembles \(\frac{x^{2}+25}{2(x + 5)}\).

Step 3: Simplifying the expression

No further simplification is possible because the numerator \(x^{2}+25\) and the denominator \(2(x + 5)\) share no common factor other than 1.

Key Concepts

Factorization in Algebra

Factorization is a method used in algebra to break down polynomials into simpler components called factors. These factors are polynomials that, when multiplied together, give back the original polynomial. The process involves finding elements that are common to each term within the polynomial, which can then be taken out (factored out) to simplify the expression.

For instance, consider the algebraic expression \(x^2 + 4x + 4\)). We look for numbers or variables that effectively divide all the terms. In this case, we notice that each term can be written as a square of \(x+2\)), leading to the factored form \(x+2)^2\)). This is what we aim for in factorization: rewriting the expression in a way that reveals its inherent structure and often simplifies further operations.

Common Factors

Identifying common factors plays a significant role in both simplifying expressions and solving algebraic equations. A common factor is a term that can divide, without a remainder, into each term of the expression. The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms.

For example, let's take the terms \(2x\)) and \(10\)). Both numbers can be divided by 2, so 2 is a common factor. In addition, since \(x\)) can be found in both terms, it is also a common factor. When dealing with more complex algebraic expressions, identifying and factoring out common factors simplifies the expressions and can make it easier to solve equations or further manipulate the algebraic expressions.

Simplification of Fractions

Simplifying fractions is a process used to reduce fractions to their simplest form. When an algebraic expression is written as a fraction, the goal is to find and cancel common factors in the numerator and the denominator. If there are no common factors other than 1, the fraction is already in its simplest form and cannot be simplified any further.

Let's consider the fraction \(\frac{x^2 + 25}{2x + 10}\)). To simplify, we would look for any common factors in the numerator and the denominator. In the example above, we were able to factor out a 2 from the denominator to get \(\frac{x^2 + 25}{2(x + 5)}\)). However, beyond this point, there are no further common factors to simplify the fraction. Therefore, we conclude that the fraction is already in its simplest form. Understanding how to simplify fractions is crucial as it allows students to work with the most streamlined version of an expression, making subsequent calculations cleaner and more straightforward.