Problem 29 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. $$ z^{2}-12 z $$

Step-by-Step Solution

Answer

Add 36 to form a perfect square trinomial: \(z^2 - 12z + 36 = (z - 6)^2\).

Step 1: Identify the Binomial Terms

The original binomial expression is given by \( z^2 - 12z \). This expression consists of a quadratic term \( z^2 \) and a linear term \( -12z \).

Step 2: Calculate the Required Constant

A perfect square trinomial takes the form \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(2ab\) corresponds to \(-12z\). Set \(2 imes b = 12\), therefore, \(b = 6\). Now, compute \(b^2\): \(b^2 = 6^2 = 36\).

Step 3: Add the Constant to Form Trinomial

Add the constant \(36\) to the binomial: \(z^2 -12z + 36\). Now the trinomial is \(z^2 -12z + 36\), which is a perfect square trinomial.

Step 4: Factor the Trinomial

The trinomial \(z^2 - 12z + 36\) can be factored as \((z - 6)^2\). This is because \((z - 6)(z - 6) = z^2 - 12z + 36\).

Key Concepts

Binomial

A binomial is a type of algebraic expression characterized by having just two distinct terms. These terms are usually joined by either addition or subtraction. In mathematics, binomials are simple yet powerful constructs that often appear in polynomial equations and require specific techniques for manipulation. - For example, in the expression \( z^2 - 12z \), we have two terms: \( z^2 \) and \( -12z \). - This example highlights a `subtraction` relationship between the terms, which is common in binomials.Binomials serve as building blocks for more complex expressions. When solving problems involving binomials, one often seeks to further simplify, expand, or transform them into another form, like a trinomial.

Trinomial

A trinomial is an algebraic expression consisting of three terms. These terms are connected by addition or subtraction. They can be seen as an extension of binomials when an extra term is added. Trinomials often appear in quadratic equations, and understanding their structure is crucial for solving higher-degree equations. - In a trinomial, you typically see a quadratic term, a linear term, and a constant, e.g., \( ax^2 + bx + c \). - For instance, the trinomial \( z^2 - 12z + 36 \) contains three terms: \( z^2 \), \(-12z \), and \( 36 \).Transforming a binomial into a perfect square trinomial, as seen in our original problem, involves adding a calculated constant to the initial two terms. This turns the expression into something that can be factored easily.

Factoring Trinomials

Factoring trinomials is a fundamental skill in algebra that involves breaking down a trinomial into a product of simpler binomials. This process can make solving equations more straightforward and helps in revealing roots or solutions. - The goal is to find two binomials whose product equals the original trinomial. - For example, \( z^2 - 12z + 36 \) can be factored as \((z - 6)^2 \), which simplifies calculations in further algebraic processes. When factoring perfect square trinomials, you often end up with a binomial squared, confirming the trinomial's status as a perfect square.

Quadratic Expressions

Quadratic expressions are prevalent in algebra and are characterized by their highest term being a squared term, i.e., \( ax^2 \). These expressions can exist as single terms, binomials, or trinomials. Understanding how to manipulate and solve quadratic expressions is essential in algebra. - They typically follow the form \( ax^2 + bx + c \) where \(a, b,\) and \(c\) are constants.Quadratic expressions play a crucial role in forming equations known as quadratic equations, which can be solved using various methods like factoring, completing the square, or using the quadratic formula. In our example, converting the binomial \(z^2 - 12z\) into the trinomial \( z^2 - 12z + 36 \) is a process of completing the square.