Problem 57 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(x^{3}+3 x^{2}+2 x+4\) for a. \(x=2\) b. \(x=-2\)

Step-by-Step Solution

Answer

For \(x = 2\), the expression evaluates to 28; for \(x = -2\), it evaluates to 4.

Step 1: Substitute x with 2

In the expression, substitute the variable x with 2. This gives: \[2^3 + 3(2)^2 + 2(2) + 4\]

Step 2: Evaluate each term for x=2

Calculate each term separately:- \(2^3 = 8\)- \(3(2)^2 = 3 \times 4 = 12\)- \(2 \times 2 = 4\)- The constant = 4

Step 3: Sum the results for x=2

Add all the evaluated terms together:\[8 + 12 + 4 + 4 = 28\]

Step 4: Substitute x with -2

In the expression, substitute the variable x with -2. This gives:\[(-2)^3 + 3(-2)^2 + 2(-2) + 4\]

Step 5: Evaluate each term for x=-2

Calculate each term separately:- \((-2)^3 = -8\)- \(3(-2)^2 = 3 \times 4 = 12\)- \(2 \times (-2) = -4\)- The constant = 4

Step 6: Sum the results for x=-2

Add all the evaluated terms together:\[-8 + 12 - 4 + 4 = 4\]

Key Concepts

Substitution Method

The substitution method is a simple and effective technique used in algebra to evaluate expressions. It involves replacing variables in an expression with specified values. This technique is particularly useful for evaluating polynomial expressions, as it allows us to calculate the expression's result for different input values. When using substitution:

• Identify the variable in the expression.
• Replace the variable with the given number.
• Perform arithmetic operations to simplify the expression and find the result.

For the expression \(x^3 + 3x^2 + 2x + 4\), we can evaluate it for any value of \(x\). For example, substituting \(x = 2\) results in each variable \(x\) being replaced by 2 throughout the expression. This method works step by step, where you first substitute, then perform operations, and finally sum up the results.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the building blocks of algebra and allow us to represent mathematical relationships and perform calculations. A typical algebraic expression can include:

• Variables such as \(x\), \(y\), etc.
• Coefficients, which are numerical factors of the terms.
• Constants, which are fixed values.
• Arithmetic operations like addition, subtraction, multiplication, and division.

For example, in the expression \(x^3 + 3x^2 + 2x + 4\), \(x\) is the variable, 3 and 2 are coefficients of \(x^2\) and \(x\) respectively, and 4 is a constant. Algebraic expressions can be evaluated at specific values using the substitution method, transforming the expression into a numerical equation that can be solved.

Polynomial Functions

Polynomial functions are special types of algebraic expressions with variables raised to non-negative integer powers. They come in various degrees, determining the highest power of the variable in the expression. A polynomial can be as simple as a constant or complex with many terms. The general form is:\[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are coefficients.
• \(x^n, x^{n-1}, \ldots, x\) are terms with non-negative exponents.

In the exercise \(x^3 + 3x^2 + 2x + 4\), it is a cubic polynomial because the highest degree is 3. Polynomial functions are essential in algebra as they can represent complex relationships and model real-world situations. They can be evaluated for specific values using substitution methods, allowing us to determine exact outputs from given inputs.