Problem 47 - Algebra 2

Question

REVIEW. The total area of a rectangle is \(25 a^{4}-16 b^{2} .\) Which factors could represent the length times width? \(\mathbf{F}\left(5 a^{2}+4 b\right)\left(5 a^{2}+4 b\right)\) \(\mathbf{G}\left(5 a^{2}+4 b\right)\left(5 a^{2}-4 b\right)\) \(\mathbf{H}(5 a-4 b)(5 a-4 b)\) \(\mathbf{J}(5 a+4 b)(5 a-4 b)\)

Step-by-Step Solution

Answer

Option G: \((5a^2 + 4b)(5a^2 - 4b)\) is the correct factorization.

Step 1: Recognize the Area Expression

The problem provides the total area of a rectangle as \(25a^4 - 16b^2\). Note that the expression \(25a^4 - 16b^2\) is a difference of squares.

Step 2: Factor the Difference of Squares

The difference of squares \(25a^4 - 16b^2\) can be factored using the formula \(x^2 - y^2 = (x + y)(x - y)\). Here, recognize that \(25a^4 = (5a^2)^2\) and \(16b^2 = (4b)^2\), so the expression can be factored as: \((5a^2 + 4b)(5a^2 - 4b)\).

Step 3: Match Factors with Given Options

Compare the factored form \((5a^2 + 4b)(5a^2 - 4b)\) with the given options. This matches exactly with Option \(\mathbf{G}\).

Step 4: Verification

Verify that multiplying \((5a^2 + 4b)(5a^2 - 4b)\) results in the original expression \(25a^4 - 16b^2\). Distribute the terms: \((5a^2)(5a^2) - (4b)(4b) = 25a^4 - 16b^2\), confirming that this is correct.

Key Concepts

Difference of Squares

When it comes to factoring polynomials, the difference of squares is an important concept to understand. It's a type of polynomial that is expressed as the subtraction of one squared term from another. The general formula is given by \( x^2 - y^2 = (x + y)(x - y) \). Here, \( x^2 \) and \( y^2 \) are perfect squares. Recognizing this pattern can help you simplify polynomial expressions dramatically.

In our exercise, the expression \( 25a^4 - 16b^2 \) fits this pattern. Notice that \( 25a^4 \) is equivalent to \( (5a^2)^2 \), and \( 16b^2 \) is \( (4b)^2 \). So applying the formula, you factor the difference of squares as: \((5a^2 + 4b)(5a^2 - 4b)\). This powerful algebraic trick allows complex problems to be broken down into simpler components that are easier to solve.

Rectangular Area

Understanding the concept of area, particularly rectangular area, is essential in algebraic expressions. The area of a rectangle is calculated by multiplying its length by its width. When given a polynomial as the area, our task is to determine the length and width represented by algebraic expressions.

In the problem provided, the total area of the rectangle is given by the expression \( 25a^4 - 16b^2 \). By factoring this difference of squares, we determine that it represents a rectangle with its length as \( (5a^2 + 4b) \) and width as \( (5a^2 - 4b) \). This is based on identifying the factors and applying the difference of squares technique.

Recognizing how polynomials relate to geometric dimensions can assist in understanding how mathematical techniques apply to real-world shapes and problems.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and involve variables and constants combined using arithmetic operations. They are versatile tools in mathematics used to represent numbers and quantities in general terms, enabling the simplification and solving of equations.

The exercise requires understanding an expression like \( 25a^4 - 16b^2 \), which initially seems complex but explains how expressions can be transformed through factoring techniques. This transformation simplifies finding relationships between terms, such as expressing a larger rectangular area in terms of its length and width.

By learning to manipulate algebraic expressions, you develop the ability to break down complex problems into more manageable parts. This is crucial for solving equations efficiently and understanding how different pieces of a problem fit together. Practicing the manipulation of algebraic expressions reinforces your skills in various math topics, building a strong foundation for further study.