Question

Factor the expression. Tell which special product factoring pattern you used. $$x^{2}+\frac{2}{3} x+\frac{1}{9}$$

Short answer

The factored expression is \((x+\frac{1}{3})^2\). The special product pattern used is the perfect square trinomial.

Step-by-step solution

Identify the Pattern

Observe the given expression and compare it with the pattern of a perfect square trinomial, which is \((a^2 + 2ab + b^2) = (a+b)^2\). Here, \(a^2=x^2\), \(2ab=\frac{2}{3}x\), and \(b^2=\frac{1}{9}\). Therefore, 'a' can be identified as the \(x\) and 'b' as \(\frac{1}{3}\).

Factorize the Expression

Replace the given expression with \((a+b)^2\) to get \((x+\frac{1}{3})^2\). This is the factorized form of the given expression.

Identify the Special Product Pattern

The factored form fits the pattern of a perfect square trinomial \((a+b)^2=a^2+2ab+b^2\), where \(a=x\) and \(b=\frac{1}{3}\).

Key concepts

Special Product Patterns

The beauty of algebra lies in the patterns that help us simplify complex expressions. One such pattern is the 'special product pattern', which includes the perfect square trinomials. A perfect square trinomial takes the form \(a^2 + 2ab + b^2 = (a + b)^2\). It's an algebraic rule that indicates a squared binomial will always expand into this specific pattern. Memorizing this can save students a great deal of effort because it allows for reverse engineering—an expression fitting this structure is immediately recognized and factored back into its binomial form.

When looking at our exercise, \(x^{2} + \frac{2}{3} x + \frac{1}{9}\), we can see it follows this rule with \(a=x\) and \(b=\frac{1}{3}\). Being aware of this pattern is crucial in algebra as it's frequently encountered in various mathematical problems, from factoring to expanding polynomials.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together represent a specific value. Expressions are the sentences of the algebraic language—they communicate mathematical ideas succinctly and precisely. A firm grasp of how to manage these expressions is essential for solving equations and understanding more complex mathematical concepts. In our example, the expression \(x^{2} + \frac{2}{3} x + \frac{1}{9}\) is made up of constant numbers, a variable (\(x\)), and involves operations of addition and exponentiation.

As students progress through algebra, they encounter expressions that grow in complexity, requiring them to apply various techniques to simplify or manipulate these expressions effectively. The exercise given is a fundamental example, illustrating how these expressions can be factored or expanded using the properties of algebra.

Factoring Techniques

Factoring is the process of breaking down a complex expression into simpler, multipliable components or 'factors'. To master algebra, one must understand different factoring techniques, which can include pulling out a greatest common factor, factoring by grouping, or utilizing special products like the difference of squares and perfect square trinomials.

In the problem at hand, knowing how to factor perfect square trinomials is the key technique. The process begins by recognizing the special product pattern, then rewriting the trinomial as the square of a binomial, \(x + \frac{1}{3})^2\). This technique is invaluable because it's not only used in isolation but also in conjunction with other methods to simplify higher-degree polynomials or to solve quadratic equations. By practising various factoring techniques, students acquire a powerful set of tools for tackling virtually any algebraic challenge.