Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Make a geometric factoring model" to represent the given factorization. For instance, a factoring model for \(2 x^{2}+5 x+2=(2 x+1)(x+2)\) is shown below.\(3 x^{2}+7 x+2=(3 x+1)(x+2)\)

Step-by-Step Solution

Answer

A geometric model for this factoring problem represents the polynomial as a large rectangle composed of smaller rectangles. The three smaller rectangles correspond to the terms in the polynomial, \(3x^{2}+7x+2\), where the length is represented by \((3x+1)\) and the breadth by \((x+2)\).

Step 1: Identify The Terms

Given the polynomial \(3x^{2}+7x+2\), which is factored as \((3x+1)(x+2)\). The terms are thus \(3x^{2}\), \(7x\), and \(2\).

Step 2: Distribute the Factors in the Parenthesis

The two parenthesis \((3x+1)\) and \((x+2)\) stand for the length and width dimensions of the entire rectangle. The whole rectangle represents \(3x^{2}+7x+2\). The rectangle is divided into three smaller rectangles with areas \((3x+1)x=(3x^{2}+x)\), \((3x+1)2=(6x+2)\) and \(x+2 = (x+2)\).

Step 3: Sketch the Geometric Model

Draw a large rectangle and divide it into three smaller rectangles. Label the length of the large rectangle with \((3x+1)\) and its width with \((x+2)\). The first small rectangle has areas \((3x^{2}+x)\), the second \((6x+2)\) and the third \(x\). Mark these areas. The sum of these three rectangles represents the entire polynomial \(3x^{2}+7x+2\).

Key Concepts

Polynomial Factorization

Polynomial factorization is an essential topic in algebra, aiding in simplifying expressions. It involves breaking down a polynomial into simpler polynomials that multiply to the original one. For example, consider the polynomial \(3x^2 + 7x + 2\). Through factorization, it becomes \((3x + 1)(x + 2)\). This means that the product of \((3x + 1)\) and \((x + 2)\) gives back the original polynomial.

• You first identify common factors or use techniques like grouping, quadratic formula, or special products.
• In the given example, the factorization reveals potential roots or solutions of the polynomial if set to zero.

Understanding factorization helps in solving equations and is particularly useful in calculus and other advanced mathematics areas.

Rectangular Representation

The rectangular representation is a visual tool used to understand polynomial factorization better. It involves creating a geometric model of the factored polynomial, representing it as a rectangle. This method helps visualize the distribution of terms.

• In our example, the factors \((3x + 1)\) and \((x + 2)\) are treated as dimensions of a rectangle.
• When multiplying these dimensions, it forms a large rectangle, equivalent in area to the original polynomial \(3x^2 + 7x + 2\).
• This rectangle is further divided into smaller rectangles each representing parts of the polynomial.

Drawing and labeling these parts on the model makes it easier to see how each term combines to form the entire polynomial, enhancing comprehension of the factorization process.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. A polynomial like \(3x^2 + 7x + 2\) is a type of algebraic expression. Understanding these expressions is crucial since they form the basis of algebra.

• An algebraic expression can include constants (numbers), variables (like \(x\)), and coefficients (numbers before variables, like 3 in \(3x^2\)).
• Operations in these expressions can include addition, subtraction, multiplication, and division.

Knowing how to manipulate algebraic expressions, through operations such as factoring or expanding, is key to solving equations and modeling real-world scenarios. This skill enables you to transform complex problems into manageable pieces, which is essential in mathematics.