Problem 67 - Elementary Algebra

Question

State the restrictions and then simplify. $$ 2 x 2-7 x-41-4 x 2 $$

Step-by-Step Solution

Answer

Simplified: \(-2x^2 - 7x - 41\). No restrictions on \(x\).

Step 1: Identify the Expression

The given expression is a quadratic expression: \(2x^2 - 7x - 41 - 4x^2\). Our task is to simplify it and state any restrictions.

Step 2: Combine Like Terms

Start by identifying and combining like terms in the expression. The expression contains two quadratic terms: \(2x^2\) and \(-4x^2\). These can be combined by performing the subtraction: \(2x^2 - 4x^2\).

Step 3: Simplify the Quadratic Terms

Subtract the quadratic terms: \(2x^2 - 4x^2 = -2x^2\). The expression now becomes \(-2x^2 - 7x - 41\).

Step 4: Check for Zero Denominator or Other Restrictions

Since the expression is a quadratic polynomial, there are no fractions involved, and no variables in denominators, so there are no restrictions on the variable \(x\).

Step 5: Final Simplified Expression

The simplified expression is \(-2x^2 - 7x - 41\). There are no restrictions on \(x\).

Key Concepts

Simplification

To simplify a mathematical expression means to condense it into a more concise and manageable form without changing its value. In the case of polynomials like the one given in our exercise, simplification involves reorganizing and reducing the expression to its simplest form. This process helps in making the expression easier to interpret and solve.

Here, the expression \( 2x^2 - 7x - 41 - 4x^2 \) can initially seem complex. But remember, the goal of simplification is to highlight its essential components by eliminating unnecessary parts or terms.

• Identify like terms that can be combined (e.g., terms involving the same power of x).
• Perform arithmetic operations to reduce the expression further (addition or subtraction as required).

By simplifying, we turn a tangled problem into a more straightforward solution, enhancing understanding and ease of use in further calculations.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions, especially polynomials. To understand this, we focus on gathering terms with identical variable components or powers. In our specific example, we have multiple terms involving \( x^2 \).

Initially, the expression is: \( 2x^2 - 7x - 41 - 4x^2 \).
The terms \( 2x^2 \) and \(-4x^2 \) are like terms because they both contain \( x^2 \) as their variable part. By subtracting these two, we combine them into: \( (2x^2 - 4x^2) = -2x^2 \).

As you can see:

• Like terms make solving much easier when combined properly.
• It’s important to pay close attention to signs (plus or minus) when combining.
• After combining like terms, check again if further simplification is possible or necessary.

Combining like terms creates a simplified expression with fewer terms, making it less complicated and easier to comprehend.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients connected by addition, subtraction, and multiplication operations. They can take many forms, from simple linear expressions to more complex quadratic or higher-degree terms. In your textbook exercise, \( 2x^2 - 7x - 41 - 4x^2 \) starts as a quadratic polynomial.

A quadratic polynomial is a special type where the highest power of the variable is squared (i.e., \( x^2 \)). This particular problem emphasizes quadratic terms, but the techniques apply to polynomials of any degree.

• Recognize the degree of the polynomial by identifying the highest power of \( x \).
• Remember that polynomials don't contain division by variables.

The solution simplifies it to a clearer form: \( -2x^2 - 7x - 41 \), showcasing that the polynomial now presents itself ready for any needed further analysis or manipulation without any restrictions on \( x \). Understanding polynomials is fundamental as they form the building blocks for more advanced algebra and calculus.