Problem 66 - Algebra 2

Question

Find \(p(7)\) and \(p(-3)\) for each function. $$ p(x)=x^{3}-11 x-4 $$

Step-by-Step Solution

Answer

\(p(7) = 262\); \(p(-3) = 2\).

Step 1: Understanding the Problem

We need to find the value of the polynomial function \( p(x) = x^3 - 11x - 4 \) for two specific inputs: \( x = 7 \) and \( x = -3 \). This involves substituting each value into the function and simplifying.

Step 1: Substitute and Simplify for x = 7

Substitute \( 7 \) into the polynomial function: \( p(7) = 7^3 - 11(7) - 4 \). Start by calculating \( 7^3 \), which is \( 343 \). Then, calculate \( 11 \times 7 = 77 \). The expression becomes \( 343 - 77 - 4 \). Simplify it: \( 343 - 77 = 266 \), and \( 266 - 4 = 262 \). Thus, \( p(7) = 262 \).

Step 2: Substitute and Simplify for x = -3

Substitute \( -3 \) into the polynomial function: \( p(-3) = (-3)^3 - 11(-3) - 4 \). Calculate \( (-3)^3 = -27 \). Then, evaluate \( 11 \times (-3) = -33 \). The expression becomes \( -27 + 33 - 4 \). Perform the addition first: \( -27 + 33 = 6 \), and then subtract \( 4 \): \( 6 - 4 = 2 \). Thus, \( p(-3) = 2 \).

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression that involves variables raised to positive integer powers. Typically, a polynomial function is written in the form of a sum:

• A variable or unknown, like \( x \).
• Coefficients, which are the numbers multiplying the variable terms.
• A set of exponents applied to the variable, indicating its power.

In our case, the polynomial function given is \( p(x) = x^3 - 11x - 4 \). Here:

• The term \( x^3 \) is the cubic term with a coefficient of 1.
• The term \( -11x \) is a linear term with the coefficient -11.
• The constant term is -4, which does not contain a variable.

Each part of the polynomial has a specific role in determining the function's behavior as \( x \) changes. The highest power (or degree) of the polynomial often gives insight into the shape of its graph, such as how it curves or if it has turning points.

Substitution Method

The substitution method is a straightforward way to evaluate functions for specific values by replacing the variable with the given number. In our original exercise, we use this method to find \( p(7) \) and \( p(-3) \).
To substitute, take the polynomial function \( p(x) = x^3 - 11x - 4 \) and replace the \( x \) with the values indicated, one at a time.

Applying Substitution

- **For \( x = 7 \):** Replace \( x \) with 7 to obtain \( p(7) = 7^3 - 11(7) - 4 \). Calculate each part step by step: first, the cube of 7, then 11 times 7, and finally subtract as needed. - **For \( x = -3 \):** Similarly, replace \( x \) with -3 yielding \( p(-3) = (-3)^3 - 11(-3) - 4 \). Again, solve each component in small steps.

Simplifying Expressions

Simplifying expressions within a polynomial is crucial for accurately evaluating them. This means combining like terms, performing arithmetic, and reducing the expression to its simplest form.
Start by calculating any exponents, such as \( x^3 \), then proceed to multiplications: here, that's \( -11x \). Finally, follow the order of operations - often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Simplification Steps

• First, handle any powers: compute \( 7^3 = 343 \) or \( (-3)^3 = -27 \).
• Next, multiply coefficients: \( 11 \times 7 = 77 \) and \( 11 \times (-3) = -33 \).
• Finally, perform the arithmetic: subtract and add what's left. Ensure each step is clear and accurate to avoid mistakes.

Simplification helps transform the polynomial into an easily comprehensible result, making it more straightforward to understand the effect of different \( x \) values on the function's output.