Problem 58 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(x^{3}-3 x^{2}-x+9\) for a. \(x=3\) b. \(x=-3\)

Step-by-Step Solution

Answer

Part (a) is 6; Part (b) is -42.

Step 1: Substitute x in Part (a)

For part (a), we substitute \(x = 3\) into the expression \(x^3 - 3x^2 - x + 9\). This gives us: \[ 3^3 - 3(3)^2 - 3 + 9. \]

Step 2: Simplify the Expression for Part (a)

Now, calculate each term: \(3^3 = 27\),\(3(3)^2 = 27\),and keep \(-3\) and \(+9\) as they are. So, the expression becomes: \[ 27 - 27 - 3 + 9. \] Simplify it to get \[ 0 - 3 + 9 = 6. \]

Step 3: Substitute x in Part (b)

For part (b), substitute \(x = -3\) into the expression. We have: \[ (-3)^3 - 3(-3)^2 - (-3) + 9. \]

Step 4: Simplify the Expression for Part (b)

Calculate each term:\((-3)^3 = -27\),\(-3(-3)^2 = -27\),and the expression becomes:\[ -27 - 27 + 3 + 9. \] Simplifying gives:\[ -54 + 3 + 9 = -42. \]

Key Concepts

Substitution Method

The substitution method is a powerful tool when evaluating polynomial expressions with a specific value for a variable. It involves replacing the variable with the given value and then simplifying the expression step by step. This method is particularly useful for verifying the results of polynomial functions by substituting different values.

To illustrate, let's consider the polynomial expression \(x^3 - 3x^2 - x + 9\). If we want to evaluate this expression for \(x = 3\), we replace every \(x\) in the polynomial with 3. The resulting expression becomes \(3^3 - 3(3)^2 - 3 + 9\).

• Make sure to correctly replace all occurrences of the variable, not missing any instance.
• Substitute with care: pay attention to negative signs or coefficients that must be maintained.

Once the substitution is complete, the next step is to simplify the expression.

Simplifying Expressions

After substitution, the next crucial step is simplifying the expression. This involves solving the arithmetic operations, often following the order of operations: parenthesis, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). This is commonly referred to as PEMDAS.

Let's break down the simplification of the expression \(3^3 - 3(3)^2 - 3 + 9\):

• First, compute the powers: \(3^3 = 27\) and \(3(3)^2 = 3 \times 9 = 27\).
• Then, substitute these values back into the expression to get \(27 - 27 - 3 + 9\).
• Finally, perform addition and subtraction in order from left to right: each operation one at a time, simplifying to the final result of 6.

Simultaneously, handle negative numbers cautiously, especially while multiplying or raising them to even powers. Effective simplification leads to an accurate evaluation of the polynomial.

Polynomial Functions

Understanding polynomial functions is key in evaluating expressions. Polynomial functions are made up of variables raised to whole number exponents and multiplied by coefficients. For example, \(x^3 - 3x^2 - x + 9\) is a third-degree polynomial function because the highest power of \(x\) is 3. Such expressions are vital in modeling various real-life situations and are the foundation for calculus and advanced algebra courses.

Key features of polynomial functions include:

• Degree: The highest power of the variable, which in this expression is 3.
• Coefficients: Numbers multiplying the variables, such as -3 for \(x^2\) and -1 for \(x\) in our expression.
• Constant term: The term in the polynomial with no variable, which here is 9.

Even when substituting specific values for \(x\), the polynomial retains these innate properties, allowing us to glean important information about its behavior and evaluate it correctly. Hence, mastering polynomial functions is crucial for successful mathematical modeling and problem-solving.