Problem 43 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. $$ z^{2}-16 z $$

Step-by-Step Solution

Answer

Add 64 to the binomial to make it a perfect square trinomial.

Step 1: Identify the Coefficient

Consider the given binomial, which is in the form \( z^2 - 16z \). Identify the coefficient of \( z \), which in this case is \(-16\).

Step 2: Calculate Half of the Coefficient

Take the coefficient of \( z \) which is \(-16\) and divide it by 2. This gives us \(-8\).

Step 3: Square the Result

Now take the result from the previous step, \(-8\), and square it. This means multiplying \(-8\) by itself: \((-8)^2 = 64\).

Step 4: Insert the Constant into the Binomial

Add the square from the previous step to the binomial: \( z^2 - 16z + 64 \). This is now a perfect square trinomial.

Step 5: Verify the Perfect Square Trinomial

Verify that the trinomial \( z^2 - 16z + 64 \) can be written as a perfect square. This expression factors as \( (z - 8)^2 \). Therefore, the trinomial is a perfect square.

Key Concepts

Understanding Binomials

A binomial is an algebraic expression that contains exactly two terms. The word "bi" typically denotes two, much like in bicycle or biped. It's a crucial building block in algebra. For example, in the expression given in the exercise, \( z^2 - 16z \), we have two terms:

• \( z^2 \), a term squared
• \( -16z \), a linear term with a negative sign

To solve problems involving binomials, it's important to identify the individual parts. Recognizing the terms in a binomial is the first step in more complex calculations, such as forming trinomials or factoring. Binomials are a foundation for understanding more advanced polynomial expressions.

Importance of Coefficients

Coefficients are the numerical part of the terms in an algebra expression. They are vital because they tell us how many times to consider each term. For instance, in \( -16z \), the coefficient is \(-16\). This reveals the multiplier of the variable \( z \).

In operations such as completing the square in the given exercise, the coefficient plays an instrumental role:

• Divide the coefficient of the linear term by 2.
• Square this result to become part of the trinomial.

Understanding coefficients aids in manipulating algebraic expressions efficiently, especially when aiming to transform a binomial into a perfect square trinomial.

Exploring Trinomials

A trinomial involves three terms. An example relates directly to our exercise, where the binomial \( z^2 - 16z \) evolved into the trinomial \( z^2 - 16z + 64 \). The basic structure usually looks like \( ax^2 + bx + c \). This structure makes trinomials a bit more complex than binomials.

To turn a binomial into a perfect square trinomial, the focus is primarily on the middle term and ensuring symmetry:

• Consider the coefficients and calculate as we did with \( -16 \).
• Ensure balanced operations that allow the trinomial to factor neatly.

Understanding the setup and transformation into trinomials is vital in algebra to solve equations effectively.

Factoring Trinomials Made Easy

Factoring trinomials means expressing them as a product of simpler expressions, usually binomials. This is a key skill when simplifying algebraic expressions or solving equations. For example, the trinomial \( z^2 - 16z + 64 \) becomes \((z - 8)^2\), signifying it's a perfect square trinomial.

The process for factoring usually involves:

• Finding two numbers whose product equals \( c \) (the constant term).
• These numbers should also sum to \( b \), the middle term coefficient.

When you successfully write the trinomial as a binomial squared, like \((z-8)^2\), you've effectively factored it. This clarity helps simplify algebraic expressions and lays the groundwork for solving equations with these expressions.