Problem 53 - Intermediate Algebra

Question

Perform the indicated operations. \(-5 x^{2}+x^{2}\)

Step-by-Step Solution

Answer

The solution is \\(-4x^2\\).

Step 1: Identify Like Terms

First, recognize that the terms \(-5x^2\) and \(x^2\) are like terms because they both contain the variable \(x^2\). Like terms have the same variable raised to the same power, making them combinable through addition or subtraction.

Step 2: Combine Like Terms

Combine the coefficients of the like terms \(-5x^2\) and \(x^2\). The operation involved is addition: \(-5x^2 + 1x^2 \), which results in \((-5 + 1)x^2\).

Step 3: Simplify the Expression

Simplify the combined expression by performing the arithmetic operation on the coefficients: \(-5 + 1 = -4\). This leads to the simplified expression \(-4x^2\).

Key Concepts

Polynomial Operations

Polynomial operations involve various actions performed on polynomials. These include addition, subtraction, multiplication, and sometimes division. When dealing with polynomial operations, it’s crucial to understand the structure of polynomials.

• **Terms**: Each polynomial consists of terms—parts of the expression separated by plus or minus signs.
• **Variables and Exponents**: Polynomials can have variables that are raised to an exponent, indicating the degree of the term.
• For example, in \(-5x^2\), \(x\) is the variable, and 2 is the exponent.
• **Coefficients**: These are the numerical parts that multiply the variable part of each term. In the term, \(-5x^2\), \(-5\) is the coefficient.

When performing polynomial operations like in the exercise, the primary action is to combine like terms—terms in the polynomial that have identical variable parts. This operation simplifies the expression, making it easier to manage and understand.

Coefficient Addition

Coefficient addition is integral to combining like terms in polynomial operations. This process involves adding the numerical coefficients of terms that are alike. Let's delve deeper into how this works in practice:

• **Identify Like Terms**: Like terms are terms that have exactly the same variable part, meaning the same variable and exponent. In our case, both \(-5x^2\) and \(x^2\) share the variable \(x^2\).
• **Add the Coefficients**: Focus on the numerical part of each like term.
  • For the terms \(-5x^2\) and \(x^2\), we have coefficients \(-5\) and \(+1\), respectively.
  • We calculate \(-5 + 1\), which gives \(-4\).

The result of this addition simplifies the polynomial by reducing it to fewer terms. This process is fundamental to managing polynomials effectively, especially when preparing for more complex algebraic manipulations.

Simplification in Algebra

Simplification in algebra is about reducing expressions to their simplest form. This process is essential to make complex problems more manageable. Here's an overview of simplification using the exercise as an example:

• **Combining Like Terms**: This is often the first step in simplification. In our example, this involved adding the coefficients of \(-5x^2\) and \(x^2\).
• **Perform Arithmetic Operations**: Calculate the arithmetic of the coefficients to simplify the expression.
  • For instance, \(-5 + 1\) simplifies to \(-4\), leading to the term \(-4x^2\).
• **Resulting Expression**: After simplifying, our final answer is a leaner, more streamlined expression, \(-4x^2\).

The simplification not only helps in solving algebraic problems efficiently but also enhances the understanding of algebraic structures. It is a powerful tool in both basic and advanced levels of mathematical problem-solving.