Problem 87 - Algebra A Combined Function

Question

Factor each expression completely. $$ x^{2}-\frac{2}{3} x+\frac{1}{9} $$

Step-by-Step Solution

Answer

The expression is factored as \((x - \frac{1}{3})^2\).

Step 1: Identify a Common Factor

First, examine the quadratic expression \( x^2 - \frac{2}{3}x + \frac{1}{9} \). Notice that there is no common factor across all terms except 1. This indicates that the expression may be a perfect square trinomial.

Step 2: Recognize the Perfect Square Form

The expression \( x^2 - \frac{2}{3}x + \frac{1}{9} \) should be checked to see if it matches the perfect square form \((a-b)^2 = a^2 - 2ab + b^2\). Here, \( a = x \) and \( b = \frac{1}{3} \).

Step 3: Rewrite the Expression as a Perfect Square

By substituting \( a = x \) and \( b = \frac{1}{3} \), we can rewrite the expression as \[ (x - \frac{1}{3})^2 = x^2 - 2 \cdot x \cdot \frac{1}{3} + (\frac{1}{3})^2 = x^2 - \frac{2}{3}x + \frac{1}{9}. \]

Step 4: Write the Final Factored Form

The expression \( x^2 - \frac{2}{3}x + \frac{1}{9} \) can be completely factored as \((x-\frac{1}{3})^2\). This is the simplest form of the expression.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is one type of quadratic expression that is derived from squaring a binomial. It takes the form \( a^2 + 2ab + b^2 \) or \( a^2 - 2ab + b^2 \) depending on the sign of the middle term. Perfect square trinomials result from expressions like \((a+b)^2\) and \((a-b)^2\).
To identify or verify a perfect square trinomial, follow these steps:

• Check if the first and last terms are squares (e.g., \( x^2 \) and \( \left(\frac{1}{3}\right)^2 \)).
• The middle term should be twice the product of the roots of the first and last terms (e.g., \(-2 \cdot x \cdot \frac{1}{3} = \frac{-2}{3}x\)).

Perfect square trinomials are important since they simplify calculations in algebra, making it easier to rewrite expressions in their condensed factored form.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest degree of the variable is two. That means these expressions commonly feature terms in the form \( ax^2 + bx + c \).
Quadratics can model many real-life situations, making them a key component of algebra.
Here are some distinctive properties to help identify them:

• The equation involves terms such as \( x^2 \) or \( y^2 \). These terms are what classify the expression as quadratic.
• The quadratic expression can be factored, solved using various methods such as factoring, completing the square, and the quadratic formula.

By mastering quadratics, students not only develop crucial problem-solving skills but can also use them across different mathematical contexts.

Factored Form

Factoring quadratic expressions means expressing them as the product of two or more simpler terms. The factored form reveals the roots of the quadratic equation and provides a clearer insight into the properties of the polynomial.
To deal efficiently with factored form:

• Recognize perfect square trinomials and patterns that match forms such as \((a+b)^2\) or \((a-b)^2\).
• Ensure the expression simplifies to factors which multiply back to give the original quadratic expression.

The factored form can make solving quadratic equations straightforward since it often directly reveals the solutions to the expression when equated to zero. Overall, factoring transforms quadratic expressions into their simplest composition, enhancing understanding and facilitating further mathematical tasks.