Problem 59 - Elementary Algebra

Question

State the restrictions and then simplify. $$ (2 x-5)(x-7)(7-x)(2 x-1) $$

Step-by-Step Solution

Answer

Restrictions: \(x \neq \frac{5}{2}, \frac{1}{2}, 7\). Expanded form: \(-(2x-5)(x^2 - 14x + 49)(2x-1)\)."}

Step 1: Identify Factors and Restrictions

The expression is given as \[ (2x-5)(x-7)(7-x)(2x-1) \]Consider each factor individually to ensure well-defined values. We note restrictions by finding values of \(x\) that make any factor zero. These restrictions are:1. \(2x - 5 = 0\), so \(x = \frac{5}{2}\).2. \(x - 7 = 0\), so \(x = 7\).3. \(7 - x = 0\), so \(x = 7\).4. \(2x - 1 = 0\), so \(x = \frac{1}{2}\).Thus, the restrictions are \(x eq \frac{5}{2}, \frac{1}{2}, 7\).

Step 2: Simplify the Expression

Notice that \((x-7)(7-x)\) can be simplified since it takes the form \((a-b)(b-a)\) which is equal to \[-(a-b)^2\].Therefore, \[(x-7)(7-x) = -(x-7)^2\].Substitute this into the expression:\[-(2x-5)(x-7)^2(2x-1)\].

Step 3: Expand and Simplify Further

First, expand \((x-7)^2\) as follows: \[(x-7)^2 = x^2 - 14x + 49\].Substitute back into the expression:\[-(2x-5)(x^2 - 14x + 49)(2x-1)\].The expression is now more challenging to simplify directly, so it's presented in this expanded form.

Key Concepts

Polynomial Simplification

Polynomial simplification involves breaking down complex polynomial expressions into a simpler, more manageable form. In the given expression, notice the presence of similar terms and structures.

An important simplification arising from the expression \((x-7)(7-x)\) is recognizing it as an example of \((a-b)(b-a)\), which simplifies to \(-(a-b)^2\). Hence, \( (x-7)(7-x)\) simplifies to \(-(x-7)^2\).

Transforming expressions like this helps streamline calculations and forms the groundwork for further simplifications in polynomial expressions. Doing so shows how even intricate polynomial expressions can be expressed more compactly and transformed into easy-to-handle forms.

Additionally, simplifying polynomials aids in understanding their behavior and can be crucial in solving polynomial equations. Practice recognizing these patterns and relationships to master simplifying polynomials.

Factoring Polynomials

Factoring polynomials is a crucial method used to break down polynomials into a product of simpler factors. Here, you often look for common patterns or identities within polynomials to manage and reduce them to simpler forms.

In the given exercise, factor expressions initially appear as \((2x-5)(x-7)(7-x)(2x-1)\). The identification of \((x-7)(7-x)\) is a key factorization. Recognizing this pattern allows you to write it as \(-(x-7)^2\), reducing the complexity of calculations.

Once simplified, factoring out expressions from such polynomials is much easier, making it possible to further expand or simplify when necessary. Learning to quickly spot these factors is essential, as it speeds up both manual and computational efforts when solving algebraic problems.

Besides simplifying, factoring plays a pivotal role in finding roots or solving equations, enabling you to express polynomials in forms that reveal key characteristics and solutions.

Algebraic Restrictions

Algebraic restrictions refer to the conditions that restrict the possible values of variables in polynomial expressions to avoid undefined operations, such as division by zero.

In the context of the given polynomial expression \((2x-5)(x-7)(7-x)(2x-1)\), algebraic restrictions are derived by setting each factor equal to zero and solving for \(x\). This ensures that the expression is defined for all allowed values.

The restrictions are:

• \(2x - 5 = 0\) gives \(x = \frac{5}{2}\);
• \(x - 7 = 0\) gives \(x = 7\);
• \(7 - x = 0\) also gives \(x = 7\).
• \(2x - 1 = 0\) gives \(x = \frac{1}{2}\).

Therefore, the expression cannot be evaluated for \(x = \frac{5}{2}, \frac{1}{2}, 7\).

By understanding these restrictions upfront, you can confidently manipulate and solve polynomial expressions without encountering undefined operations, vital for ensuring the accuracy of your solutions.