Problem 61 - Algebra Form and Function

Question

Write an expression in \(x\) representing the result of the given operations on \(x\). Is the expression linear in \(x ?\) Add \(5,\) multiply by \(x,\) subtract \(2 .\)

Step-by-Step Solution

Answer

Answer: No, the resulting expression is not linear in x.

Step 1: Add 5 to x

First, we need to add 5 to \(x\). This gives us the expression \(x + 5\).

Step 2: Multiply the result by x

Next, we need to multiply the expression from step 1 by \(x\). So, we have \((x + 5) \cdot x\). Distributing the \(x\), we get the expression \(x^2 + 5x\).

Step 3: Subtract 2 from the product

Now, we need to subtract 2 from the expression in step 2. This gives us the final expression \(x^2 + 5x - 2\).

Step 4: Check if the expression is linear

Lastly, we want to check if the expression \(x^2 + 5x - 2\) is linear in \(x\). To be linear, the expression must have a constant term, a term with a power of \(x^1\), and no higher powers. In our case, there is a term with a power of \(x^2\), specifically \(x^2\). This means that the expression is not linear in \(x\). So, the resulting expression after performing the operations on \(x\) is \(x^2 + 5x - 2\), and it is not linear in \(x\).

Key Concepts

Linear Expressions

Linear expressions are one of the most foundational concepts in algebra. They represent the simplest form of polynomial expression where the variable involved is only raised to the power of one. An expression like \(ax + b\), where \(a\) and \(b\) are constants, is an example of a linear expression.
Linear expressions take the form of a straight line when graphed on a coordinate plane.
Their general characteristic features include:

• They consist of one variable raised to the first power (e.g., \(x\)).
• They can have one or more terms, but none will involve a variable raised to a power higher than one.
• The graph of a linear expression is always a straight line.

Understanding linear expressions are crucial since they form the basis for exploring more complex algebraic topics, like systems of equations and linear functions. In this example, the presence of \(x^2\) in the expression \(x^2 + 5x - 2\) makes it non-linear.

Polynomial Functions

Polynomial functions are algebraic expressions composed of variables and coefficients arranged in terms of powers. In general, a polynomial can be expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(n\) is a non-negative integer.
Each term consists of a coefficient (\(a\)) and its corresponding variable (\(x\)) raised to a whole-number power.

• They can have one or more terms.
• The highest degree of the variable dictates the degree of the polynomial (e.g., \(x^2\) in \(x^2 + 5x - 2\) is a second-degree polynomial).
• Polynomial functions are continuous and smooth curves when graphed.

In the exercise, \(x^2 + 5x - 2\) is a polynomial with three distinct terms: a quadratic term \(x^2\), a linear term \(5x\), and a constant \(-2\). Understanding polynomial functions allows you to solve a wide variety of algebraic problems, from factoring to calculus.

Operations on Variables

Operations on variables are fundamental techniques utilized in generating and simplifying algebraic expressions. They involve arithmetic operations such as addition, subtraction, multiplication, and division applied to variables.

• **Addition:** Increases the value of the variable or constant by a certain amount.
• **Subtraction:** Decreases the value of the variable or constant by a given amount.
• **Multiplication:** Involves scaling the variable or expression by a factor.
• **Division:** Involves separating the variable or expression into equal parts.

In our exercise, we explore these operations in sequence on \(x\):

• Start by adding 5, yielding \(x + 5\).
• Followed by multiplying by \(x\), which distributes to \((x + 5) \cdot x = x^2 + 5x\).
• Finally, subtract 2 to achieve the expression \(x^2 + 5x - 2\).

Mastering these operations enables one to manipulate and transform expressions effectively, a critical skill in algebra.