Question

Factor the expression. Tell which special product factoring pattern you used. $$y^{2}+12 y+36$$

Short answer

The factored expression is \((y+6)^2\). The special product factoring pattern used is a perfect square trinomial.

Step-by-step solution

Identify the Perfect Square Trinomial

The general form of a perfect square trinomial is \(a^2 + 2ab + b^2\), which can be factored into \((a + b)^2\). Look at the expression \(y^{2}+12 y+36\), you can identify that it follows this pattern because \(y^2\) is the square of 'y', 36 is the square of 6, and 2xy is equivalent to 12y for x = y and y = 6.

Factorize the expression

Following the perfect square trinomial factoring pattern \((a+b)^2\), replace 'a' with 'y' (because 'y' is the square root of \(y^2\)) and 'b' with '6' (because 6 is the square root of 36). This gives you \((y+6)^2\).

State the Factoring Pattern Used

In this case, a perfect square trinomial factoring pattern was used, which is written generally as \((a+b)^2\) or \((a-b)^2\).

Key concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables, represented by symbols like 'x', 'y', or 'z', stand in for unknown values that can be determined through various algebraic processes. For instance, in the given exercise, the expression \(y^{2}+12 y+36\) is a quadratic algebraic expression because it has a term with a variable raised to the second power (the \(y^{2}\) term). When attempting to understand such expressions, identifying each term and potential factors is crucial to simplifying and solving the equation.

Key elements of an algebraic expression include coefficients (numerical factors of terms, like the 12 in \(12 y\)), constants (like the 36), and the aforementioned variables. To simplify expressions or solve equations, we often look for patterns or common factors, one such routine involving the factorization process, especially when dealing with quadratic expressions.

Factoring Patterns in Algebra

Factoring patterns are shortcuts in algebra that help simplify expressions and solve equations more efficiently. These patterns represent frequently occurring scenarios that, once recognized, can be dealt with methodically. Think of them as the math equivalent of recognizing a word by sight rather than by sounding out each letter.

Common factoring patterns include:

• Difference of Squares: \(a^2 - b^2 = (a + b)(a - b)\)
• Perfect Square Trinomials: \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 - 2ab + b^2 = (a - b)^2\)
• Sum or Difference of Cubes: \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)

Identifying the correct pattern is like unlocking a puzzle—the right key opens the door with ease. In our exercise, the recognition of a perfect square trinomial allows us to quickly rewrite the expression in its factored form.

Perfect Square Trinomial

The concept of a perfect square trinomial is a subset of factoring patterns where the trinomial can be expressed as the square of a binomial. A trinomial is simply an algebraic expression with three terms. A 'perfect square' in this context means the expression can be broken down into a binomial that is multiplied by itself. Recognizing a perfect square trinomial is a valuable skill because it simplifies what could potentially be a complex factorization process.

Identifying a Perfect Square Trinomial

The general form of a perfect square trinomial is \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), and it factors into \(a + b)\)^2 or \(a - b)^2\) respectively. Confirm that the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

The exercise provided presents us with \(y^{2} + 12 y + 36\), which is a textbook example of a perfect square trinomial because \(y^{2}\) is a perfect square of 'y', and 36 is a perfect square of 6. The middle term, 12y, is exactly twice the product of 'y' and 6. This recognition allows us to employ the pattern to write the factored form as \(y + 6)^2\).