Problem 66 - Elementary and Intermediate Algebra

Question

Evaluate. $$ 1+x^{3}, \text { for } x=-2 $$

Step-by-Step Solution

Answer

The expression evaluates to -7 when x=-2.

Step 1 - Substitute the Value of x

Begin by substituting the given value of x, which is -2, into the expression. So substitute -2 for x in the expression 1 + x^3.

Step 2 - Evaluate the Cubic Term

Next, evaluate the cubic term \(x^3\) with x = -2. This means you need to find \((-2)^3\).

Step 3 - Simplify the Cubic Term

Calculate \((-2)^3\). This equals -8 because \(-2 \times -2 \times -2\) results in -8.

Step 4 - Add the Simplified Cubic Term to 1

Add the simplified cubic term (-8) to 1. So, the expression becomes 1 + (-8).

Step 5 - Simplify the Final Expression

Combine the numbers: 1 + (-8) = -7.

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing a variable with a given number to evaluate an expression. For example, in the expression \(1 + x^3\), we need to substitute \(-2\) for \(x\).

This means whenever we see \(x\) in the expression, we replace it with \(-2\). This substitution helps us to transform the algebraic expression into a numerical one, which is easier to solve.

For an expression \(1 + x^3\) with \(x = -2\), substitution changes it to \(1 + (-2)^3\). This step is crucial because it sets up everything needed for evaluation.

Cubic Term Evaluation

Evaluating a cubic term means raising a number to the power of three. For instance, \(x^3\) refers to multiplying \(x\) by itself twice more. When \(x = -2\), we need to calculate \((-2)^3\).

Here's the step-by-step process:

• First, multiply \(-2\) by itself: \(-2 \times -2 = 4\)
• Then, multiply the result by \(-2\) again: \(-2 \times 4 = -8\)

This gives us \((-2)^3 = -8\). The negative sign is important because an odd power of a negative number remains negative.

Expression Simplification

Expression simplification involves combining like terms and performing arithmetic operations. After evaluating the cubic term \((-2)^3\) to get \(-8\), we replace \(x^3\) in the expression \(1 + (-2)^3\) with \(-8\).

So, it becomes \(1 + (-8)\). Simplify this by combining the two numbers:

• 1 + (-8) equals -7 because adding a negative number is the same as subtracting it.

Thus, the simplified expression \(1 + x^3\) for \(x = -2\) is \-7\. Simplification is the final step to get a clear numerical result from the algebraic expression.