Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for each value of \(x\). (If not possible, state the reason.)\(-x^{3}+2 x-1 \quad\) (a) \(x=0 \quad\) (b) \(x=2\)

Step-by-Step Solution

Answer

For \(x = 0\), the expression is simplified to -1, while for \(x = 2\), the expression is simplified to -3.

Step 1: Substitute \(x = 0\) into the expression

Replace x in the expression \(-x^{3}+2 x-1\) with 0. This will give us \(-0^{3}+2*0-1\).

Step 2: Simplify after substituting \(x = 0\)

Perform the mathematical operations that emerged after the substitution. This will result in the value: -1.

Step 3: Substitute \(x = 2\) into the expression

Replace x in the expression \(-x^{3}+2 x-1\) with 2. This will give us \(-2^{3}+2*2-1\).

Step 4: Simplify after substituting \(x = 2\)

Perform the mathematical operations that emerged after the substitution. This will result in the value: -3.

Key Concepts

Substitution Method

The substitution method is a technique used to evaluate polynomial expressions by replacing the variable with a specific value. In this context, we have the polynomial expression \(-x^{3} + 2x - 1\). To find the value for specific inputs, we simply substitute the given x-value into the expression.

Here’s how you can efficiently apply substitution for the expression given:

• Identify the expression to evaluate. In this case, \(-x^{3} + 2x - 1\).
• Substitute the chosen value of \(x\) into the expression. For example, for \(x = 0\) and \(x = 2\).
• After substitution, simplify the new numerical expression to obtain the result.

Substitution is straightforward and is fundamental when dealing with polynomials, since it allows us to determine specific outcomes depending on different inputs.

Simplification Process

Simplification in mathematics involves reducing expressions to their simplest form. After substituting a value into a polynomial expression, simplification is the next crucial step. For our given expression \(-x^{3} + 2x - 1\), simplification involves carrying out arithmetic operations that arise after substitution.

Consider these steps:

• After substitution, execute exponentiation first. For instance, \(-2^{3}\) becomes \(-8\).
• Then proceed with multiplication. Multiply the coefficients by the substituted values, such as \(2 imes 2 = 4\).
• Finally, perform addition or subtraction as required. Combine the outcomes to reach a simplified result.

This process makes complex expressions more manageable and ensures accurate evaluation. It highlights how elegantly polynomials can be manipulated towards simpler, clearer results.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables raised to whole number exponents and their coefficients. They can be as simple as a single term or as complex as having multiple terms. Our initial expression, \(-x^{3} + 2x - 1\), is a polynomial of degree 3.

Polynomials are crucial in both theoretical and applied mathematics. They help model and solve real-world problems, such as calculating areas, predicting profits, or analyzing scientific data.

• Monomials consist of a single term.
• Binomials have two terms.
• Polynomials with three terms, such as our expression, are called trinomials.

By understanding how to evaluate these expressions using the substitution method and simplification process, students can better appreciate the power and versatility of polynomial mathematics.