Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each algebraic expression for the given value of the variable. $$3 x^{2}-8 x ; x=-2$$

Step-by-Step Solution

Answer

The evaluated result of the algebraic expression, \(3x^2 - 8x\), for \(x = -2\) is \(28\).

Step 1: Substitution

Replace \(x\) in the algebraic expression \(3x^2 - 8x\) with \(x = -2\). This gives us \(3(-2)^2 - 8(-2)\)

Step 2: Simplification

First, calculate the value of \((-2)^2\), which equals \(4\), and substitute it back into the expression, which gives us: \(3 * 4 - 8(-2)\). Next, multiply \(3 * 4 = 12\) and \(-8*(-2) = 16\). This simplifies the new expression to \(12 + 16\).

Step 3: Final Calculation

Now, calculate the final expression: \(12 + 16 = 28\).

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept in algebra that involves replacing a variable with a given number. This process helps in evaluating expressions, making it a key skill for solving algebraic problems.
In the exercise, we were given the expression \(3x^2 - 8x\) and asked to evaluate it for \(x = -2\). By substituting \(x = -2\) into the expression, we transform it into a numerical form: \(3(-2)^2 - 8(-2)\).

• The essence of substitution is straightforward, yet it is crucial to carefully handle any negative values or fractions.
• Ensure all occurrences of the variable in the expression are replaced consistently.

By doing this correctly, you set up the expression for further evaluation and simplification.

Polynomial Evaluation

Polynomial evaluation involves computing the value of a polynomial expression once a specific value for the variable is provided.
The aim is to determine the numerical outcome of the expression through arithmetic calculations.
For our specific example:

• Start with the calculation of powers and products. Here, compute \((-2)^2\), resulting in \(4\).
• This transforms our expression to \(3 \times 4 - 8 \times (-2)\).

Next, resolve the products:

• Multiply \(3\) and \(4\) to get \(12\).
• Calculate \(-8 \times (-2)\) to receive \(16\).

This step-by-step breakdown is essential in polynomial evaluation as it sets the stage for simplifying the expression further.

Expression Simplification

Simplifying an expression means combining like terms and following arithmetic rules to reduce the expression to its simplest form.
This is the final stage where all calculation parts come together.
After evaluating the individual parts of the polynomial, we get \(12 + 16\).

• By adding these two results, we conclude with \(28\) as the final answer.

Expression simplification requires attention to detail, ensuring all arithmetic operations, such as addition, subtraction, multiplication, or division, are executed correctly. It leads to the most compact and clear form of the expression.