Problem 45 - Algebra for College Students

Question

Factor completely. $$x(x+2)+5(x+2)$$

Step-by-Step Solution

Answer

The fully factored form is \((x+2)(x+5)\).

Step 1: Identify Common Factor

The expression given is \( x(x+2) + 5(x+2) \). Notice that both terms \( x(x+2) \) and \( 5(x+2) \) share a common factor, which is \((x+2)\). Factoring out the common factor simplifies the expression.

Step 2: Factor Out the Common Factor

Factor \((x+2)\) out from both terms of the expression:\[ (x+2)(x) + (x+2)(5) = (x+2)(x+5) \]. Now, the expression is fully factored.

Key Concepts

Common Factor

In algebra, identifying the common factor is a key step in simplifying and factoring expressions. A common factor refers to a term or value that is common across all parts of an expression. It is much like finding the commonality in a group of items, which simplifies solving. The process of spotting common factors is crucial in making complex expressions more manageable.

To identify a common factor, look for a shared term across the entire equation. This can be numbers, variables, or a group/combination as seen in polynomial expressions. For example, in the expression \( x(x+2) + 5(x+2) \), both terms include \((x+2)\). This group \((x+2)\) is the common factor. Recognizing these patterns can simplify the factors and reduce errors.

• Always scan for numbers or variables that repeat across terms.
• Consider the entire expression to identify shared units.
• Think of factoring as a way to "undress" an expression to see its simplest form.

Factoring Expressions

Factoring expressions is the procedure of breaking down a complex expression into simpler components or 'factors'. It's a bit like unpacking a box into smaller, easily understandable parts. This is particularly useful in algebra where equations can become quite complex. The purpose is to rewrite the expression as a product of simpler factors.

In our example \( x(x+2) + 5(x+2) \), we already detected \((x+2)\) as a common factor. Now, the process involves 'factoring out' this shared component. This means you will rewrite the expression by pulling the \((x+2)\) out of each term, resulting in:

• Original Expression: \( x(x+2) + 5(x+2) \)
• Factored Form = \((x+2)(x+5)\)

This transformation makes the equation much more navigable. It also sets up additional algebraic maneuvers, such as solving for variables or finding roots.

Polynomial Simplification

Simplifying a polynomial is the art of making the expression as straightforward as possible. It involves identifying patterns and using algebraic rules to collapse the polynomial into its simplest terms. Once factors are identified and extracted, the polynomial is significantly easier to work with.

For our expression \( x(x+2) + 5(x+2) \), by factoring out \((x+2)\) the polynomial is simplified. This leads to a cleaner, more compact expression \((x+2)(x+5)\), which is fully factored. Simplifying expressions like this helps in reading equations, solving them, and predicting their behavior.

• Unclutter complex expressions by recognizing repeating factors.
• Once simplified, expressions are more flexible for further algebraic work.
• Key goal of simplification is not just aesthetic, it enhances understanding and eases computations.