Problem 23 In Exercises $23-30$, factor t... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 23

In Exercises $23-30$, factor the trinomial. $$ x^{2}-7 x z-18 z^{2} $$

Short Answer

Expert verified

The factorization of the trinomial $x^{2}-7 x z-18 z^{2}$ is $(x - 2z)(x + 9z)$

Step by step solution

Set up Factoring

The trinomial should be factored into two binomials, each of the form $(x + a)(x - b)$ where $a * b = -18z^2$ and $a - b = -7z$. Hence, the task is to find numbers $a$ and $b$ that meet these conditions.

Determine the values of 'a' and 'b'

Upon examination, the numbers that satisfy these conditions are -2 and 9. This is because -2 * 9 = -18, and -2 - 9 = -7. Hence, 'a' = -2 and 'b' = 9.

Write down the factored form

Plug 'a' and 'b' into the binomials to get the factored form of the trinomial. Using the obtained 'a' and 'b' values, the trinomial $x^{2}-7 x z-18 z^{2}$ can be factored as $(x - 2z)(x + 9z)$. This is the required factorization of the given trinomial.

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Key Concepts

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

Understanding Binomials

A binomial is an algebraic expression that contains two terms. For example, in the expression $(x - 2z)$, we see two distinct terms: $x$ and $-2z$. These terms are separated by a subtraction or addition sign. Binomials are simple yet powerful components in algebra. They often appear in factoring problems as recipients for division or multiplication by other expressions. Here, the simple structure of a binomial makes it a building block for more complex polynomials. When we factor trinomials, we frequently aim to express them as a product of two binomials.

Exploring Trinomials

Trinomials are polynomials composed of three terms. The exercise we are tackling here involves factoring a trinomial:

$x^2$
$-7xz$
$-18z^2$

These terms come together to create what we call a quadratic trinomial. This is a specific type of polynomial where the highest degree is two. The process of factoring such trinomials is aimed at rewriting them as a simple product of two binomials. Recognizing the structure of a trinomial is crucial because it guides how we approach the problem of simplifying it.

Basics of Algebra Factoring Methods

Factoring is a fundamental algebraic skill where we rewrite expressions as products of their factors.
The trinomial in this exercise is factored using a trial-and-error method. This method involves identifying two numbers that multiply to give the constant term of the trinomial and add up to the coefficient of the middle term. Here, the constant term is $-18z^2$ and the middle term coefficient is $-7z$. We find these numbers (in this case, -2 and 9) after considering several combinations, then proceed to use them in forming our binomials: $(x - 2z)$ and $(x + 9z)$.

This strategic choice breaks down complex expressions into manageable parts.
Factoring simplifies problems, making equations easier to solve or further manipulate.
It's all about recognizing patterns and applying the right rules to suit each problem.

With the correct application of factoring methods, algebra becomes less intimidating and more intuitive.

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91影视

In Exercises \(23-30\), factor the trinomial. $$ x^{2}-7 x z-18 z^{2} $$

Short Answer

Step by step solution

Set up Factoring

Determine the values of 'a' and 'b'

Write down the factored form

Key Concepts

Understanding Binomials

Exploring Trinomials

Basics of Algebra Factoring Methods

One App. One Place for Learning.

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