Problem 14 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ x^{2}-12 x+36 $$

Step-by-Step Solution

Answer

The trinomial factors as \((x - 6)^2\).

Step 1: Identify a perfect square relationship

The trinomial is of the form \(x^2 - 12x + 36\). Observe that the first term \(x^2\) is a perfect square. The last term, \(36\), is also a perfect square since \(6^2 = 36\). Furthermore, check whether the middle term (-12x) matches the pattern \(-2ab\) where \(a = x\) and \(b = 6\). Indeed, \(-2 \cdot x \cdot 6 = -12x\). This indicates a perfect square trinomial.

Step 2: Write as a squared binomial

Since the trinomial follows the perfect square pattern, it can be written as a binomial squared. Thus, \(x^2 - 12x + 36\) can be expressed as \((x - 6)^2\).

Step 3: Verify the factorization

Expand \((x - 6)^2\) to ensure it matches the original trinomial. \((x - 6)^2 = (x - 6)(x - 6) = x^2 - 6x - 6x + 36 = x^2 - 12x + 36\). Since the expanded form matches the original trinomial, \((x - 6)^2\) is the correct factorization.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of trinomial that arises when a binomial is squared. In essence, this type of polynomial takes the form \( a^2 \pm 2ab + b^2 \), where \( a \) and \( b \) are real numbers or algebraic expressions.

The key to identifying a perfect square trinomial is recognizing the pattern:

• The first and last terms must be perfect squares themselves.
• The middle term must equal \( \pm 2ab \), meaning it must be twice the product of the square roots of the first and last terms.

For instance, in the trinomial \( x^2 - 12x + 36 \):

• \( x^2 \) is a perfect square, as \( (x)^2 = x^2 \).
• \( 36 \) is a perfect square, as \( (6)^2 = 36 \).
• The middle term, \(-12x\), fits the expression \(-2 \cdot x \cdot 6 \), confirming the pattern.

Thus, recognizing this relationship helps one easily factor the trinomial into a binomial square.

Binomial Squared

A binomial squared takes the form \((a \pm b)^2\). This translates into \(a^2 \pm 2ab + b^2\) when expanded. This concept simplifies the process of factoring trinomials considerably, especially when the trinomial is recognized as a perfect square.

Let’s consider the example \( (x - 6)^2 \). Here:

• \( a = x \)
• \( b = 6 \)

The squared binomial then expands like this:

• \((x - 6)(x - 6) \)
• \(= x^2 - 6x - 6x + 36 \)
• Combine like terms to get \(x^2 - 12x + 36 \)

This step by step verification reaffirms our factorization and showcases how efficient recognizing a binomial squared can be.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They form the building blocks for various mathematical equations and problems, allowing for complex calculations and solutions.

When dealing with algebraic expressions, especially trinomials like \( x^2 - 12x + 36 \), being able to factor them into simpler forms such as binomials can greatly simplify problems you're solving.

• Expressions can be manipulated by using mathematical operations.
• Simplification, like factoring, helps in solving equations or understanding functions.

In the context of the original exercise, transforming the expression into \((x - 6)^2\), makes working with the trinomial much easier in solving further equations or applications. Factoring is an essential skill in algebra that allows these expressions to be broken down into manageable parts, ultimately helping solve or simplify larger algebraic tasks.