Problem 90 - Introductory Algebra for College Students

Question

Solve each equation .Use a calculator to help with the arithmetic. Check your solution using the calculator. Evaluate \(x^{3}-4 x\) for \(x=-1 .\)

Step-by-Step Solution

Answer

The evaluated value of the equation \(x^{3}-4 x\) for \(x=-1\) is \(3\).

Step 1: Substitution

Start by substituting \(-1\) for \(x\) in the equation. The equation now reads as \((-1)^{3}-4(-1)\)

Step 2: Evaluation

Next, calculate the value of the equation with the substituted value. The computation becomes \(-1 + 4\).

Step 3: Solution

The simplified answer is \(3\).

Key Concepts

Substitution Method

The substitution method is a common algebraic technique useful for simplifying and solving equations. In this approach, you replace the variable with a specific value. This helps you evaluate the expression or solve the equation.

Think of substitution as swapping out a placeholder for a specific number to see how it behaves in a mathematical context. In our given exercise, the variable \(x\) was replaced by \(-1\). By doing this, the equation \(x^3 - 4x\) turned into \((-1)^3 - 4(-1)\).

Steps to perform substitution:

• Identify the variable you need to substitute.
• Replace the variable with the given value in the equation.
• Proceed with solving or simplifying the resulting expression.

Substitution is straightforward and is particularly useful when checking the correctness of a solution by plugging back into the original equation.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial for a specific value of its variable. A polynomial is a mathematical expression consisting of variables and coefficients, with powers that are whole numbers.

Here, we evaluated the polynomial \(x^3 - 4x\) for \(x = -1\). This means calculating the expression's value after substituting \(x\) with \(-1\).

Steps for evaluating a polynomial:

• Substitute the given value into the polynomial in place of the variable.
• Follow the order of operations: parentheses, exponents, multiplication, division, addition, and subtraction (PEMDAS).
• Simplify the expression to find the resultant value.

In this exercise, the substitution gave us \[(-1)^3 - 4(-1) = -1 + 4 = 3.\] Polynomial evaluation is essential for understanding how variables interact with each other in an equation.

Calculator Usage

Using a calculator can be an essential tool, especially when dealing with complex arithmetic during algebraic evaluations. While it's important to understand the underlying math, calculators help verify your calculations and ensure accuracy.

When using calculators for polynomial expressions like \(x^3 - 4x\):

• First, perform any substitutions manually or use the calculator to do so.
• Input the expression exactly as intended, using parentheses where necessary to maintain order of operations.
• Check that the calculator's output matches your manual calculations to confirm your result.

In this particular exercise, a calculator was used to assist and verify that the polynomial evaluates to \(3\) when \(x\) equals \(-1\). This verification step ensures that your manual calculations were accurate, providing confidence in your skill and understanding of the material.