Problem 117 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I factored \(9-25 x^{2}\) as \((3+5 x)(3-5 x)\) and then applied the commutative property to rewrite the factorization as \((5 x+3)(5 x-3)\)

Step-by-Step Solution

Answer

The statement does not make sense because the correct re-arrangement utilizing the commutative property would result in \((5x+3)(3-5x)\), not \((5 x+3)(5 x-3)\).

Step 1: Check the Initial Factoring

First, let's analyze the correctness of the initial factorization. \(9-25 x^{2}\) should factor into \((3-5x)(3+5x)\) because it's difference of squares. Here, a is 3 and b is 5x in the formula \(a^2 - b^2 = (a-b)(a+b)\). After comparing the given factorization \((3+5 x)(3-5 x)\) with the correct factorization, it's clear that they are equivalent due to the commutative property of addition.

Step 2: Verify the Application of the Commutative Property

Next, let's examine the resulting factorization \((5 x+3)(5 x-3)\) that is supposed to occur by applying the commutative property, which states that elements can be rearranged without affecting the result. By re-arranging terms within each binomial, \((3+5 x)\) becomes \((5x+3)\) and \((3-5 x)\) becomes \((-5x+3)\), or equivalently \((3-5x)\). So, the resulting factorization is not \((5x+3)(5x-3)\) but should be \((5x+3)(3-5x)\). Therefore, the given statement 'does not make sense'.

Key Concepts

Difference of Squares

The concept of "Difference of Squares" is a fundamental rule in algebra that simplifies polynomial expressions. It states that any expression of the form \(a^2-b^2\) can be rewritten as \((a-b)(a+b)\). This occurs because the middle terms cancel each other out when you distribute or foil the product \((a+b)(a-b)\). For example, if we look at the expression \(9 - 25x^2\), we can identify it as a difference of squares because \(9\) is \(3^2\) and \(25x^2\) is \((5x)^2\).

• Use the formula \(a^2-b^2 = (a-b)(a+b)\) to factor expressions.
• The squares must be perfect squares, meaning they are results of a number multiplied by itself.
• Be careful with negative signs; the structure always involves a subtraction, qualifying it as a 'difference' of squares.

Always watch out for the key elements that identify a difference of squares, whether terms are in the correct format, and ensure you've picked the right values during your factoring process.

Commutative Property

The Commutative Property is a basic principle in mathematics, dealing with the order of operation in addition and multiplication. In simple terms, it means that the order in which you add or multiply numbers does not change the result. For instance, in addition, \(a + b = b + a\), and similarly, in multiplication, \(a \cdot b = b \cdot a\).

When it comes to factoring polynomials, the commutative property allows us to rearrange the terms without impacting the end value. In the example of factoring \(9-25x^2\) as \((3+5x)(3-5x)\), applying the commutative property involved changing \((3+5x)\) to \((5x+3)\). This doesn't affect the essence of the expression but does highlight that we must still factor correctly first, aligning terms with their intended order.

• Rearranging terms in addition or plant terms in a polynomial does not affect the product due to the commutative property.
• This property does not apply to subtraction or other operations that are not commutative.

Understanding this property ensures that mistakes in arrangement do not lead to errors in more complex algebra problems.

Polynomial Factoring Mistakes

Factoring polynomials involves breaking down a polynomial into simpler components, but mistakes can occur if the principles are not fully understood. Common mistakes include misapplying rules such as the difference of squares or mishandling signs during operations. A typical error might involve reversing terms incorrectly or forgetting to apply the commutative property properly.

In the exercise example, the mistake was assuming that rearranging using the commutative property would yield a valid factorization. Instead, the statement presented an incorrect final factorization.

• Always ensure the initial factorization is correct.
• Be careful with negative signs, especially in subtraction.
• Check each factorization step thoroughly.

Thoroughly verify each aspect of your work to avoid jumping to conclusions based on misunderstandings of properties like commutative property or algebraic rules.