Problem 34 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{x^{2}+5 x}{(x+1)^{2}}+\frac{4}{(x+1)^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+4}{x+1}\).

Step 1: Combine the Fractions

Since both fractions have the same denominator \((x+1)^2\), we can combine them by adding their numerators. This gives us: \(\frac{x^2 + 5x + 4}{(x+1)^2}\).

Step 2: Factor the Numerator

Next, we attempt to factor the numerator \(x^2 + 5x + 4\). It factors to \((x+1)(x+4)\).

Step 3: Simplify the Expression

The expression \(\frac{(x+1)(x+4)}{(x+1)^2}\) can be simplified by canceling the common factor \((x+1)\) in the numerator and the denominator. This simplifies to \(\frac{x+4}{x+1}\).

Key Concepts

Adding Fractions

When you're dealing with adding fractions, the key is to make sure that they have the same denominator. The denominator is the bottom part of a fraction. In your problem, the fractions already have a common denominator, \((x+1)^2\), so you can add them more easily. By combining terms with a common denominator, what you actually do is keep the denominator the same and add the numerators, the top parts, together.

Here's how it works in simpler terms:

• Check if both fractions have the same denominator. If they do, great! If not, you would typically find a common denominator, but that's already done here.
• Add the numerators together. For the problem, you take \(x^2 + 5x\) from the first fraction and add it to \(4\) from the second fraction, resulting in \(x^2 + 5x + 4\).

By doing so, you've successfully combined the fractions into a single fraction with a simplified numerator. This method of adding doesn't change the size or value of the original fractions, just represents them in a combined form.

Factoring Polynomials

Factoring is like breaking numbers or expressions down into simpler pieces called "factors" that can be multiplied to give you back the original number or expression. Polynomials, which are expressions consisting of variables and coefficients, can often be factored. Here, the polynomial \(x^2 + 5x + 4\) in the numerator needs factoring.

Factoring polynomials typically involves finding two numbers that multiply to give you the last term (here it's \(4\)) and add to give you the middle coefficient (here it's \(5\)). For this specific polynomial:

• Find numbers that multiply to \(4\) and add up to \(5\). These numbers are \(1\) and \(4\).
• Write the polynomial in its factored form, which is \((x+1)(x+4)\).

Understanding factoring is crucial because it can reveal common factors that can be simplified in rational expressions. Once factored, you can more easily spot these common elements, which leads us into the next concept: canceling common factors.

Canceling Common Factors

After factoring, the next step is simplifying the expression. Simplification often involves canceling out common factors in both the numerator and the denominator of a fraction. In the polynomial equation, \((x+1)(x+4)\) is the numerator and \((x+1)^2\) is the denominator.

Here's how you cancel common factors:

• Identify any terms or factors that appear in both the numerator and the denominator. In this exercise, \((x+1)\) is a common factor.
• Cancel these common factors by dividing them out of both the numerator and the denominator. This leaves you with \(\frac{(x+4)}{(x+1)}\).

Canceling common factors simplifies the expression and makes it easier to work with. It's important to only cancel factors that are multiplied in the numerator and denominator, not added or substracted. This ensures the equivalence of the simplified expression to the original.