Problem 51
Question
For the following exercises, consider this scenario: Charlotte has appointed chairperson to lead a city beautification project. The first actis to install statues and fountains in one of the city's parks. The park is a rectangle with an area of \(98 x^{2}+105 x-27 \mathrm{m}^{2},\) as shown in the following figure. The length and width of the park are perfect factors of the area. Factor by grouping to find the length and width of the park.
Step-by-Step Solution
Verified Answer
The length and width of the park are \( (14x - 3) \) and \( (7x + 9) \).
1Step 1: Identify the polynomial
The area of the park is given by the polynomial \(98x^2 + 105x - 27\). Our task is to factor this polynomial to find the dimensions of the park, which are its length and width.
2Step 2: Group the terms
To factor by grouping, start by splitting the middle term, \(105x\), so that we can group parts of the polynomial in pairs: \[98x^2 + 126x - 21x - 27\].
3Step 3: Factor out the common factors
Look at each group from the expression in step 2:- From \(98x^2 + 126x\), factor out \(14x\): \(14x(7x + 9)\).- From \(-21x - 27\), factor out \(-3\): \(-3(7x + 9)\).
4Step 4: Write as a product of common binomial factors
Now notice that both groups contain a common factor of \(7x + 9\). Rewrite the expression as:\[(14x - 3)(7x + 9)\].
5Step 5: Determine the dimensions
The factors found, \(14x - 3\) and \(7x + 9\), represent the possible dimensions of the park: one is the length, and the other is the width.
Key Concepts
AlgebraRectangular AreaFactor by GroupingPolynomial Expressions
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. These symbols represent quantities without fixed values, known as variables.
In the context of solving problems like finding the dimensions of a park's area given by a polynomial, algebra involves the use of expressions, equations, and techniques such as factoring to find solutions.
A polynomial, like the one we see in this problem, is an algebraic expression consisting of terms whose variables have non-negative integer exponents, like the terms in the polynomial expression of the park's area: \(98x^2 + 105x - 27\).
The primary goal here is to re-write a polynomial into a product of simpler polynomials, which gives insight into the values of the variables involved.
In the context of solving problems like finding the dimensions of a park's area given by a polynomial, algebra involves the use of expressions, equations, and techniques such as factoring to find solutions.
A polynomial, like the one we see in this problem, is an algebraic expression consisting of terms whose variables have non-negative integer exponents, like the terms in the polynomial expression of the park's area: \(98x^2 + 105x - 27\).
The primary goal here is to re-write a polynomial into a product of simpler polynomials, which gives insight into the values of the variables involved.
Rectangular Area
When dealing with problems involving rectangular areas, it's important to understand how area works. The area of a rectangle can be calculated by multiplying its length by its width, which are its two dimensions.
Given a polynomial representing an area, like in this exercise, by factoring it, you unfold the dimensions of the rectangle.
Rectangles are fundamental shapes often studied in geometry and algebra. They're used to model spaces in real-world scenarios, such as parks or rooms. When tasked with finding dimensions, understanding this relationship between area and its contributing factors is crucial.
Given a polynomial representing an area, like in this exercise, by factoring it, you unfold the dimensions of the rectangle.
Rectangles are fundamental shapes often studied in geometry and algebra. They're used to model spaces in real-world scenarios, such as parks or rooms. When tasked with finding dimensions, understanding this relationship between area and its contributing factors is crucial.
Factor by Grouping
Factoring by grouping is a strategy used to factor polynomials that have four terms. This technique is particularly useful when other methods of factoring do not readily apply.
This method involves three main steps: splitting the polynomial into sections, factoring out common factors from each section, and then uniting the sections using a shared binomial factor.
In our exercise, we first split the middle term of the polynomial \(105x\) into two parts, allowing us to form groups: \(98x^2 + 126x - 21x - 27\).
Next, we factored each group separately, looking for common factors, before combining them into \((14x - 3)(7x + 9)\), the factors of our original polynomial.
This method involves three main steps: splitting the polynomial into sections, factoring out common factors from each section, and then uniting the sections using a shared binomial factor.
In our exercise, we first split the middle term of the polynomial \(105x\) into two parts, allowing us to form groups: \(98x^2 + 126x - 21x - 27\).
Next, we factored each group separately, looking for common factors, before combining them into \((14x - 3)(7x + 9)\), the factors of our original polynomial.
Polynomial Expressions
Polynomial expressions consist of terms made up of coefficients, variables, and exponents. A typical polynomial is a sum of one or more terms, such as the expression \(98x^2 + 105x - 27\) in this example.
Polynomials are a central focus in algebra because they represent a wide range of phenomena, from simple quadratic equations in school mathematics to complex equations modeling real-world scenarios.
Understanding polynomial expressions involves grasping how to manipulate these terms to simplify or rewrite expressions. Techniques such as factoring are key to working successfully with polynomials.
In this exercise, finding the polynomial factors involved recognizing the structure of a polynomial and applying strategic methods like grouping to break it down into its basic components.
Polynomials are a central focus in algebra because they represent a wide range of phenomena, from simple quadratic equations in school mathematics to complex equations modeling real-world scenarios.
Understanding polynomial expressions involves grasping how to manipulate these terms to simplify or rewrite expressions. Techniques such as factoring are key to working successfully with polynomials.
In this exercise, finding the polynomial factors involved recognizing the structure of a polynomial and applying strategic methods like grouping to break it down into its basic components.
Other exercises in this chapter
Problem 50
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