Problem 67 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$1.8 x^{2}-5.1 x^{2}+4.1 x^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(0.8x^2\).

Step 1: Identify Like Terms

Examine the polynomial expression to identify terms that are similar. In the given expression \(1.8x^2 - 5.1x^2 + 4.1x^2\), all the terms are like terms since they have the same variable part \(x^2\).

Step 2: Combine Coefficients of Like Terms

Add the coefficients of the like terms together. This process involves adding and subtracting the coefficients in the expression:\[1.8 - 5.1 + 4.1\]The power of \(x\) remains the same because we are only combining the numerical coefficients.

Step 3: Calculate the Numerical Sum

Perform the arithmetic operation to simplify the expression:\[1.8 - 5.1 + 4.1 = 0.8\]

Step 4: Write the Simplified Expression

Combine the calculated coefficient with the common variable part:\(x^2\). Thus, the simplified expression is:\[0.8x^2\]

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a core skill in mathematics, often needed to solve problems more efficiently. When we talk about simplification, we're focusing on reducing an expression to its simplest form by combining like terms and performing arithmetic operations.

In algebra, like terms are those that have exactly the same variable raised to the same power, commonly found in expressions that involve polynomials. For instance, in the expression \(1.8x^2 - 5.1x^2 + 4.1x^2\), all terms contain \(x^2\), making them like terms. This shared characteristic is what allows us to combine them.

To simplify such an expression, you need to:

• Identify the like terms, which have the same variable and exponent.
• Combine these terms by adding or subtracting their coefficients.

It's important to note that the variables themselves do not change when combining. Only the numerical coefficients are affected. Applying these steps effectively makes the expression much easier to work with later in more complex mathematical equations.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients, involving the operations of addition, subtraction, and multiplication. What sets polynomials apart is the structure, which does not include division by a variable.

These expressions can have multiple terms. Each term in a polynomial is composed of a constant multiplied by variables raised to whole number powers. For instance, in the expression \(1.8x^2 - 5.1x^2 + 4.1x^2\), each term is formed by a coefficient and the common power of \(x^2\).

Polynomials are foundational in mathematics due to their versatility and application in various areas such as calculus, physics, and engineering. Understanding how to manipulate and simplify polynomial expressions is crucial, as it leads to more advanced concepts and real-world applications.

A crucial part of working with polynomial expressions is ensuring that they are in their simplest form through simplification. This often involves combining like terms and ensuring that all parts of the expression are aligned according to algebraic rules.

Arithmetic Operations

Arithmetic operations are fundamental in mathematics and involve basic operations: addition, subtraction, multiplication, and division. In the context of simplifying algebraic expressions like polynomials, addition and subtraction are frequently applied to combine like terms.

When combining like terms, it's essential to focus on the coefficients of the terms. In our example, \(1.8 - 5.1 + 4.1\), we're performing a series of arithmetic operations exclusively on these coefficients. The variable, \(x^2\), remains unaffected during these operations because polynomial simplification strictly pertains to handling numerical parts separately from the variable parts.

Throughout your math journey, you'll repeatedly apply these arithmetic operations in different scenarios, each time building on your ability to assess and manipulate not just basic numbers, but more complex algebraic terms effectively. Having a strong grasp of arithmetic operations makes other algebraic processes much easier and more intuitive to handle.