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Chapter 8: Problem 29

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

Short Answer

Expert verified
The missing constant is 36, and the trinomial factors to \( (z - 6)^2 \).

Step by step solution

01

Identify 'a' from the binomial

The binomial provided is in the form \( z^2 - 12z \). For a perfect square trinomial, we use the formula \( (z - a)^2 = z^2 - 2az + a^2 \). Here, \( 2a = 12 \), so we solve for \( a \): \( a = \frac{12}{2} = 6 \).
02

Calculate the missing constant

To complete the square, we need to add \( a^2 \) to the expression. From the previous step, we found \( a = 6 \), therefore \( a^2 = 6^2 = 36 \). So, the constant to add is 36.
03

Form the perfect square trinomial

Add the found constant 36 to the binomial: \( z^2 - 12z + 36 \). This transforms our expression into a perfect square trinomial.
04

Factor the trinomial

Recognize the trinomial as a perfect square of the form \( (z - a)^2 \). Substitute the value \( a = 6 \) to rewrite the trinomial: \( (z - 6)^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Trinomials
Factoring trinomials is like finding the pieces of a puzzle that fit together perfectly. In a trinomial, you have three terms that, when factored, reveal a simpler expression. To factor a trinomial, especially a perfect square trinomial, you need to identify constant terms and understand how they relate to each other. For example, in the trinomial \( z^2 - 12z + 36 \), the idea is to transform it into a form that reveals its components as a squared binomial expression, like \((z-6)^2 \).

Understanding the process:
  • Identify a pattern: Look for a squared term and a linear term where the middle term is twice the product of the square root of the first and last terms.
  • Add the correct constant: Determine which constant is needed to make the first two terms fit this recognizable pattern and ultimately form a perfect square trinomial.

By recognizing these patterns, you factor trinomials more easily, which is crucial in simplifying polynomial expressions.
Completing the Square
Completing the square is a method used to solve quadratic equations and rewrite quadratic expressions in a more useful form. Think of it as re-arranging terms to create a perfect square trinomial. This makes the equation more manageable and often easier to solve or analyze, especially when it comes to graphing.

Steps to complete the square include:
  • Ensure the equation is solely in terms of one variable, like \( z^2 - 12z \).
  • Identify the coefficient of the linear term and divide it by two. This value is crucial for both identifying what completes the square and factoring it later.
  • Add the square of this new value to both sides of the equation to maintain equality. This transforms the expression into a perfect square trinomial.

Using completing the square, you transform \( z^2 - 12z \) by adding 36, creating a trinomial \( z^2 - 12z + 36 \), which simplifies to \((z - 6)^2 \). It鈥檚 an essential technique in algebra, especially for solving quadratic equations when factoring isn鈥檛 straightforward.
Binomial Expressions
A binomial expression consists of two terms, like \( z^2 \) and \(- 12z \) as seen in the exercise. Binomials are fundamental building blocks in algebra; they form the basis for creating trinomials, which can often be factored back into binomials if they're perfect square trinomials.

The relationship between binomials and trinomials involves:
  • Combining two terms correctly: To make a binomial that can be turned into a perfect square trinomial, the terms need to be related such that one is the square of a single variable term, and the other involves the product of this variable with another constant.
  • Completing to form a perfect square: By adding the correct constant, you turn a simple binomial like \( z^2 - 12z \) into a recognizably perfect square trinomial \((z - 6)^2 \).

Ultimately, understanding binomial expressions helps simplify and solve more complex algebraic expressions by showing how they can be re-expressed in expanded or factored form, paving the way for solving equations efficiently.

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Most popular questions from this chapter

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