Problem 67 - Algebra and Trigonometry

Question

Factor the expression completely. $$ x^{3}+2 x^{2}+x $$

Step-by-Step Solution

Answer

The expression is factored as \(x(x + 1)^2\).

Step 1: Identify Common Factors

First, look for common factors in all the terms of the expression. The given expression is \(x^3 + 2x^2 + x\). Notice that each term contains \(x\) as a factor. Factoring \(x\) out of the entire expression gives: \[ x(x^2 + 2x + 1) \]

Step 2: Factor the Quadratic

Now, focus on the quadratic expression within the parentheses, \(x^2 + 2x + 1\). We need to factor this further. To do this, find two numbers that multiply to \(1\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers are \(1\) and \(1\). Rewrite the quadratic as: \( (x + 1)(x + 1) \) or simply \( (x + 1)^2 \).

Step 3: Combine the Factors

Combine the common factor found in the first step with the factors of the quadratic. The completely factored form of the expression is: \[ x(x + 1)^2 \]

Key Concepts

Common Factor

When solving polynomial expressions, the first step often involves identifying and factoring out the Common Factor. Imagine you have several terms within an expression, much like an assortment of colorful beads on a string. The Common Factor is like finding the groove that all the beads share. Discovering this factor is critical, as it simplifies the expression and makes further operations straightforward.

For example, in the expression \(x^3 + 2x^2 + x\), the Common Factor is \(x\). This is because each term contains \(x\) as a multiplier. By extracting this factor, we reduce the complexity, thereby rewriting the expression as \(x(x^2 + 2x + 1)\).

Remember:

• Identify the greatest common factor in all terms.
• Factor it out to simplify the expression.

Spotting the Common Factor is a foundational skill in algebra that lays the groundwork for more complex operations.

Quadratic Expression

A Quadratic Expression is often symbolized by variables raised to a power of two. Its canonical form looks like \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions appear frequently in all aspects of algebra and mathematics.

Consider the quadratic part of our initially reduced expression: \(x^2 + 2x + 1\). Here,

• \(a = 1\)
• \(b = 2\)
• \(c = 1\)

The role of these constants dictates the shape and position of the parabola if graphed.

Understanding the structure of a Quadratic Expression aids in visualization and problem-solving, making it easier to spot patterns and symmetries. Knowing how to decompose these expressions is also an essential step before proceeding to factor them.

Factoring Quadratics

Factoring Quadratics is the next pivotal step after simplifying an expression by identifying a common factor. This process involves breaking down a quadratic expression into simpler, multiplied factors.

Consider the quadratic from the expression: \(x^2 + 2x + 1\). To factor it:

• Find two numbers that multiply to the constant term, \(c\), which is 1.
• The same numbers must add up to the middle coefficient, \(b\), which is 2.

These magic numbers are both 1. Thus, the expression can be rewritten as \((x + 1)(x + 1)\) or conveniently, \((x + 1)^2\).

Mastering this art of factoring allows for efficient problem-solving in algebra, making complex equations more manageable.