Problem 42 - Algebra Form and Function

Question

Is the given expression linear in the indicated variable? Assume all constants are non-zero. $$ 2 a x+b x+c, x $$

Step-by-Step Solution

Answer

Answer: Yes, the expression $$2ax + bx + c$$ is linear in the variable \(x\).

Step 1: Identify the terms in the expression

The given expression is $$2ax + bx + c$$. We can see that there are three terms in this expression: 1. \(2ax\) 2. \(bx\) 3. \(c\)

Step 2: Determine the degree of \(x\) in each term

Now we will determine the degree of the variable \(x\) in each term. 1. In the term \(2ax\), the degree of \(x\) is 1, as \(x\) is raised to the power of 1 (multiplied by itself once). So, this term is linear in \(x\). 2. In the term \(bx\), the degree of \(x\) is also 1, as the variable \(x\) is raised to the power of 1. Therefore, this term is linear in \(x\) as well. 3. In the term \(c\), there is no variable \(x\) present. It is a constant term.

Step 3: Check for higher-degree terms and conclude

As we have identified the degree of \(x\) in each term, we can see that there are no terms with a higher degree than 1 in terms of \(x\). The highest degree of \(x\) is 1, and there are no higher-degree terms. Hence, the given expression $$2ax + bx + c$$ is linear in the variable \(x\).

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is key to identifying the nature of the expression. The degree is determined by the highest power to which the variable in the expression is raised.
For any polynomial, the degree provides crucial information:

• If an expression is linear, as in our exercise, the degree is 1. This means the variable is only raised to the first power, like in terms such as \(2ax\) and \(bx\).
• If there was a term where the variable was squared (e.g., \(x^2\)), the degree of the polynomial would be 2, making it a quadratic expression.
• Higher-degree polynomials are named differently (cubic for degree 3, quartic for degree 4, etc.), but in a linear expression, the highest degree is never more than 1.

This concept helps determine the kind of relationship the expression represents, with linear often implying a straight-line relationship in graphical terms.

Variables in Algebra

Variables are symbols used to represent unknown or changeable values in algebraic expressions. In algebra, the variable allows us to generalize mathematical relationships.
A few key points about variables in algebra:

• Variables like \(x\) can take any value in a set, often representing real numbers.
• They are placeholders that can help create formulas or functions to solve for unknowns.
• Understanding their role is crucial because most operations and solutions in algebra revolve around manipulating these symbols to solve equations or simplify expressions.

In our examined expression \(2ax + bx + c\), \(x\) is the variable of interest. Here, it allows us to determine whether the expression is linear by checking the power to which \(x\) is raised.

Constant Terms

Constant terms are numbers on their own in an expression, meaning they do not multiply by variables or change when variables change.
Some distinct characteristics of constant terms include:

• They are fixed values like \(c\) in our expression \(2ax + bx + c\), offering no change when substituting different values of \(x\).
• Constant terms contribute to the value of an expression but do not influence the variable's power or the degree of the polynomial.
• In expressions and equations, constants are crucial for forming equations because they provide a defined numerical element.

Understanding constant terms helps when evaluating and simplifying expressions, as they represent the parts of an expression that remain unchanged.