Problem 35 - College Algebra with Modeling and Visualization

Question

Subtract the polynomials. $$(5 x-3)-(2 x+4)$$

Step-by-Step Solution

Answer

The result is \(3x - 7\).

Step 1: Understand the Expression

The given polynomials to subtract are \((5x - 3)\) and \((2x + 4)\). Our task is to subtract the second polynomial from the first.

Step 2: Distribute the Negative Sign

Distribute the negative sign across the polynomial being subtracted: \((5x - 3) - (2x + 4) = 5x - 3 - 2x - 4\).

Step 3: Combine Like Terms

Combine like terms by grouping coefficients of \(x\) and constant terms: For the \(x\) terms: \(5x - 2x = 3x\).For the constant terms: \(-3 - 4 = -7\).

Step 4: Write the Result

Combine the simplified terms to write the final expression: The result is \(3x - 7\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It involves multiplying a single term by each term within a polynomial. In our example, subtraction is happening, which is much like distributing a negative one across the terms of the second polynomial.
In the expression \[(5x - 3) - (2x + 4)\]you distribute the negative sign across \[(2x + 4)\], treating it as if multiplying by \(-1\).
This means the expression transforms into:

• \(5x - 3 - 2x - 4\)

Through this property, each term in the subsequencing polynomial changes sign. The crucial part is to ensure you apply this properly, making sure the subtraction affects each component within the brackets.

Combining Like Terms

Combining like terms simplifies expressions by adding or subtracting coefficients for terms that share the same variable. In our expression\[5x - 3 - 2x - 4\],we identify like terms to simplify.

• The 'like terms' involving \(x\) are \(5x\) and \(-2x\). Combining these, we calculate: \(5x - 2x = 3x\).
• The constant terms are \(-3\) and \(-4\). Adding these gives us: \(-3 - 4 = -7\).

When simplifying, ensure all variable terms are grouped and simplified appropriately, and do the same for all constant terms. Grouping terms appropriately is key because it leads to the most concise and simplified form of your expression.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols. In our task, we deal with two algebraic expressions: \[5x - 3\]and\[2x + 4\].
When subtracting these expressions, observe how each component interacts through the operation. Variables and constants are organized separately:

• Variable terms are terms that include letters, such as \(5x\) and \(2x\).
• Constant terms are standalone numbers, like \(-3\) and \(4\).

Understanding the structure of algebraic expressions enables you to apply operations accurately and efficiently. Recognizing parts of the expression and knowing how they can be manipulated is crucial for mastering algebra.