Problem 20 - Algebra 2

Question

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=2 x^{3}-x^{2}+5 x-7\)

Step-by-Step Solution

Answer

\( p(4) = 125 \), \( p(-2) = -37 \).

Step 1: Understand the Problem

We need to evaluate the polynomial function \( p(x) = 2x^3 - x^2 + 5x - 7 \) at two different points: \( x = 4 \) and \( x = -2 \). This means we will substitute 4 and -2 into the function and calculate the result.

Step 2: Evaluate p(x) at x = 4

Substitute \( x = 4 \) into the polynomial expression: \[ p(4) = 2(4)^3 - (4)^2 + 5(4) - 7 \] Calculate each term: - \( 2(4)^3 = 2 \times 64 = 128 \)- \( (4)^2 = 16 \)- \( 5 \times 4 = 20 \)- Constant term: -7Combine all terms:\[ p(4) = 128 - 16 + 20 - 7 = 125 \]

Step 3: Evaluate p(x) at x = -2

Substitute \( x = -2 \) into the polynomial expression:\[ p(-2) = 2(-2)^3 - (-2)^2 + 5(-2) - 7 \] Calculate each term: - \( 2(-2)^3 = 2 \times -8 = -16 \)- \( (-2)^2 = 4 \)- \( 5 \times (-2) = -10 \)- Constant term: -7 Combine all terms:\[ p(-2) = -16 - 4 - 10 - 7 = -37 \]

Step 4: Verify the Results

Check each calculation and step to ensure there are no mistakes. We used substitution properly, and arithmetic operations were checked for errors. The evaluations have been done correctly, resulting in \( p(4) = 125 \) and \( p(-2) = -37 \).

Key Concepts

Substitution Method

The substitution method is a technique used to evaluate expressions by plugging specific values into a given mathematical formula. When evaluating polynomial functions, this method becomes essential since we substitute the variable with specific numbers to simplify the expression and find the solution. For example, if we want to find the value of a polynomial function at specific points like in our exercise when substituting for \(x = 4\) or \(x = -2\), each occurrence of \(x\) in the expression is replaced. The steps to follow with substitution are straightforward:

• Identify the value to substitute for the variable.
• Replace each occurrence of the variable with this value in the expression.
• Carry out the appropriate arithmetic operations to simplify the expression.

By following these steps, the substitution method simplifies what could otherwise be complex expressions into a basic arithmetic problem, making it easier to evaluate.

Arithmetic Operations

Arithmetic operations are foundational mathematical processes that include addition, subtraction, multiplication, and division. When evaluating polynomial functions, the substitution will lead you to require using these operations to simplify and resolve expressions.Understanding the order of operations is crucial, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that every calculation is conducted correctly. For instance, within a polynomial like \(2x^3 - x^2 + 5x - 7\), careful attention must be paid to:

• Calculating exponents before moving to multiplication.
• Carrying out all multiplication and division operations before addition and subtraction.

Making sure arithmetic operations are performed accurately ensures the correct evaluation of expressions, which is particularly important in more complex polynomials.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of terms that involve powers of a variable. The general form is \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(n\) is a non-negative integer, and the coefficients \(a_n, a_{n-1}, ..., a_0\) are constants.These functions are characterized by:

• Having one or more terms.
• The highest power of the variable determines the degree of the polynomial.
• Smooth, continuous graphs without any breaks.

Understanding polynomial functions helps in identifying each part of the expression, and how they contribute to the overall function value. It's important to note that polynomial functions are the building blocks for more complex computations in algebra and calculus.

Evaluating Expressions

Evaluating expressions is the process of determining the value of an algebraic expression by substituting the variables with specific values and simplifying the result. To successfully evaluate an expression, such as a polynomial, involves:

• Substituting each variable with the given number.
• Performing arithmetic operations in the correct order.
• Summarizing all parts to find the final value.

In our example problem, evaluating \(p(x)\) required substituting \(4\) and \(-2\) into the polynomial function, performing calculations, and combining like terms. This execution results in numeric values that represent the function's behavior at those specific points, giving insight into the nature and graph of the function.