Problem 3 - Algebra 2

Question

Find p(3) and p(-1) for each function. \(p(x)=-x^{3}+x^{2}-x\)

Step-by-Step Solution

Answer

\(p(3) = -21\) and \(p(-1) = 3\).

Step 1: Substitute x = 3 into p(x)

To find \(p(3)\), substitute \(x = 3\) into the function: \(p(x) = -x^3 + x^2 - x\). This gives \(p(3) = -(3)^3 + (3)^2 - 3\).

Step 2: Calculate p(3) Step by Step

Calculate each term: 1. \(-(3)^3 = -(27) = -27\).2. \((3)^2 = 9\).3. \(-3 = -3\).Add these results together: \(-27 + 9 - 3 = -21\). Thus, \(p(3) = -21\).

Step 3: Substitute x = -1 into p(x)

To find \(p(-1)\), substitute \(x = -1\) into the function: \(p(x) = -x^3 + x^2 - x\). This gives \(p(-1) = -(-1)^3 + (-1)^2 - (-1)\).

Step 4: Calculate p(-1) Step by Step

Calculate each term: 1. \(-(-1)^3 = 1\).2. \((-1)^2 = 1\).3. \(-(-1) = 1\).Add these results together: \(1 + 1 + 1 = 3\). Thus, \(p(-1) = 3\).

Key Concepts

Substitution Method

The substitution method is a crucial technique for evaluating polynomial functions, which is particularly helpful for solving algebraic expressions. When we want to find the value of a polynomial at a specific point, say \( x = a \), we use substitution. This means we replace every instance of \( x \) in the polynomial with the number \( a \). Doing this allows us to determine the value of the polynomial function at that particular \( x \)-value with straightforward calculations. Let's break down this process a little further:

• Identify the polynomial function you need to work with.
• Choose the specific value that will replace \( x \); this value is usually given in the problem.
• Substitute this value directly into the function by replacing every \( x \) with the chosen number.
• Calculate based on the order of operations: exponents first, then multiplication, and finally addition or subtraction.

Substitution is essential in algebra because it provides a concrete number as an output for a given input, simplifying many algebra problems.

Polynomial Functions

Polynomial functions form the backbone of algebra and calculus due to their wide applicability and simple structure. A polynomial function is defined as a mathematical expression consisting of variables raised to whole number powers and coefficients that are combined using addition, subtraction, or multiplication. For example, \( p(x) = -x^3 + x^2 - x \) is a polynomial function.Polynomial functions are characterized by:

• The "degree," which is the highest power of the variable \( x \) in the expression. For the function \( -x^3 + x^2 - x \), the degree is 3.
• "Coefficients" which are the numbers that multiply the variables, such as -1, 1 in our example function.
• "Terms," which are the separate elements joined by addition or subtraction in the polynomial, like \(-x^3\), \(x^2\), and \(-x\).

Understanding polynomial functions allows for easier manipulation and simplification of algebraic expressions, and they appear frequently in equations modeling real-world scenarios.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They represent values and can be simplified or solved through various algebraic methods, such as the substitution method discussed earlier. In algebra, expressions are built using a mixture of constants, variables, such as \( x \), and operations like addition, subtraction, multiplication, and division.Key points to remember about algebraic expressions:

• They do not have an equality sign, unlike equations, and are treated as subjects of operations.
• They serve as the base for constructing more complex mathematical equations and functions.
• Substitution can help evaluate these expressions at particular values of the variables, providing numeric results for otherwise abstract entities.
• Expressions can be simplified using common mathematical properties and rules such as combining like terms and using distributive properties.

Algebraic expressions play a critical role in mathematical problem-solving and in expressing formulas used in various scientific fields.