Problem 54 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{2 x+x^{2}}{4-x^{2}}$$

Step-by-Step Solution

Answer

Fraction does not simplify further.

Step 1: Factor the Numerator

Look at the numerator, which is \(2x + x^2\). Factor out the common term, which is \(x\): \[x(x + 2)\].

Step 2: Factor the Denominator

Observe the denominator, \(4 - x^2\). This is a difference of squares, which can be factored as: \[(2 - x)(2 + x)\].

Step 3: Write the Fraction with Factored Terms

Rewrite the fraction \(\frac{2x + x^2}{4 - x^2}\) using the factored numerator and denominator: \[\frac{x(x + 2)}{(2 - x)(2 + x)}\].

Step 4: Simplify the Fraction by Canceling Common Terms

There are no common factors between the numerator and the denominator to cancel out directly. However, note that \((x + 2) = -(2 + x)\), so we cannot directly cancel these unless we rearrange to match the factor, which isn't needed here as it leads to an improper matching with negatives.

Key Concepts

Factoring Polynomials

Polynomials are expressions that consist of variables and coefficients, organized as a sum of terms. Factoring polynomials involves rewriting them as a product of their simpler factors. In our example, the numerator is the polynomial \(2x + x^2\). To factor it, we first look for the greatest common factor (GCF) among the terms. Here, both terms share \(x\) as a common factor. By pulling \(x\) out, we simplify the expression to \(x(x + 2)\). This process breaks down the polynomial into simpler parts, which can be useful for simplifying or solving equations. Factoring polynomials is crucial because it allows us to see and cancel common terms when simplifying fractions.

Difference of Squares

The difference of squares is a specific type of polynomial expression, defined as \(a^2 - b^2\). This expression can always be factored into two binomials: \((a - b)(a + b)\). In our exercise, the denominator \(4 - x^2\) is a difference of squares. Here, \(4\) is \(2^2\) and \(x^2\) is obviously \(x\) squared, allowing us to rewrite the expression as \((2 - x)(2 + x)\).

• The formula helps to easily break down such expressions, making it easier to manipulate them in algebra.
• Recognizing the difference of squares quickly is useful for factoring and simplifying fractions.

When you see an expression like \(a^2 - b^2\), always remember this factoring trick to simplify your work.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form by dividing both the numerator and the denominator by their greatest common factor or cancelling common terms. In our example, after factoring, the fraction becomes \(\frac{x(x + 2)}{(2 - x)(2 + x)}\). Normally, you would look for terms that appear in both the numerator and the denominator to cancel them out.
However, in this case, there is nothing that can be directly canceled, since \((x + 2)\) corresponds to \(-(2 + x)\) due to differences in order and sign. Consequently, it indicates the expression is already in its simplest form, emphasizing the need to be careful about signs and expressions in algebraic simplification.
Through practice, you'll become proficient at quickly discerning which terms can be simplified, making calculations more efficient and comprehensible.