Problem 63 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{3}+2 x^{2}+x $$

Step-by-Step Solution

Answer

The expression factors to \( x(x+1)^2 \).

Step 1: Identify Common Factors

The expression is \( x^3 + 2x^2 + x \). First, identify any common factors in all the terms. Notice that each term has at least one \( x \), so we factor out \( x \) from the entire expression.

Step 2: Factor Out the Greatest Common Factor

Factor \( x \) out from each term in the expression. This gives:\[x(x^2 + 2x + 1)\]

Step 3: Recognize and Factor the Quadratic Expression

Now, take a closer look at the quadratic expression inside the parentheses: \( x^2 + 2x + 1 \). This can be factored further. Recognize this as a perfect square trinomial because it matches the pattern \( a^2 + 2ab + b^2 \). Here, \( a = x \) and \( b = 1 \), which means you can rewrite it as \((x+1)^2\).

Step 4: Combine Factored Terms

Substitute the factored quadratic back into the expression. The completely factored expression becomes:\[x(x+1)^2\]

Key Concepts

Common Factors

When solving polynomial expressions, the first step often involves identifying common factors. These are elements shared by all terms in the expression. In the given problem, the expression is \( x^3 + 2x^2 + x \). Each term in this expression contains the variable \( x \). Thus, \( x \) is a common factor. This means \( x \) is a part you can "pull out" or factor from each term to simplify the expression.

When you factor out the \( x \), you're left with \( x(x^2 + 2x + 1) \). Factoring not only simplifies the expression but also makes subsequent steps easier. It's like finding a common thread that runs through all parts of the expression.

• Original Expression: \( x^3 + 2x^2 + x \)
• Common Factor Identified: \( x \)
• Factored Form: \( x(x^2 + 2x + 1) \)

Recognizing common factors is the foundation of simplifying and solving polynomial equations.

Quadratic Expression

After factoring out the common \( x \) from the polynomial, the next step is to focus on the quadratic expression that's left: \( x^2 + 2x + 1 \). A quadratic expression is a second-degree polynomial, which means the highest exponent of the variable is 2.

Quadratic expressions take the general form of \( ax^2 + bx + c \). In our case, it is \( 1x^2 + 2x + 1 \). Recognizing the structure of a quadratic expression is crucial because these can often be factored further, revealing more about the polynomial's characteristics and simplifying the problem.

• Form of Quadratic: \( ax^2 + bx + c \)
• Expression in Problem: \( x^2 + 2x + 1 \)

Understanding quadratics and how they can be factored or solved using the quadratic formula, completing the square, or other methods is a central part of algebra.

Perfect Square Trinomial

A perfect square trinomial is a specific form of quadratic expression where the expression can be written as the square of a binomial. It follows the pattern \( a^2 + 2ab + b^2 = (a+b)^2 \).

In the expression \( x^2 + 2x + 1 \), you can observe that it fits this pattern where \( a = x \) and \( b = 1 \). This means that \( x^2 + 2x + 1 \) is indeed a perfect square trinomial. Thus, it can be rewritten as \((x+1)^2\).

• Pattern: \( a^2 + 2ab + b^2 \)
• Expression: \( x^2 + 2x + 1 \)
• Factored Form: \((x+1)^2\)

Recognizing perfect square trinomials simplifies problems significantly, allowing you to express complex quadratic terms in a more compact and interpretable binomial square format. This is a powerful algebraic tool that highlights the beauty and symmetry in polynomial expressions.