Problem 11 - Algebra A Combined Function

Question

If \(P(x)=x^{3}+2 x-3\) and \(Q(x)=7 x+5,\) find each function value. $$ P(-2) $$

Step-by-Step Solution

Answer

The value of \(P(-2)\) is \(-15\).

Step 1: Understand the problem

We have two polynomial functions, \(P(x) = x^3 + 2x - 3\) and \(Q(x) = 7x + 5\). The task is to find the value of the function \(P(x)\) when \(x = -2\).

Step 2: Substitute x in P(x)

Substitute \(x = -2\) into the function \(P(x)\): \(P(-2) = (-2)^3 + 2(-2) - 3\).

Step 3: Calculate the cube

Calculate \((-2)^3\). Since the negative base raised to an odd power remains negative, \((-2)^3 = -8\).

Step 4: Multiply and simplify other terms

Next, calculate \(2(-2)= -4\). Then the expression becomes \(P(-2) = -8 - 4 - 3\).

Step 5: Perform the remaining operations

Add the terms together: \(-8 - 4 - 3 = -15\). Therefore, \(P(-2) = -15\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. When you have a polynomial function like \(P(x) = x^3 + 2x - 3\), evaluating the function means substituting the variable \(x\) with a given number and performing the arithmetic operations. Here’s why this process is straightforward and essential:

• **Substitute the Variable**: Replace \(x\) with the specific number you're evaluating. In our case, it's \(-2\).
• **Perform Operations**: Carefully follow algebraic rules to compute the expressions. Recall the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• **Interpret the Result**: Once you've calculated, the resulting number is the value of the function at that point. For \(P(-2)\), the result was -15.

Evaluating functions is a fundamental skill in algebra that applies to various types of functions beyond just polynomials. Mastering this skill will help in understanding how functions behave and are used in different mathematical contexts.

Polynomial Equations

Polynomial equations are expressions that set polynomials equal to a certain value, often zero, such as \(P(x) = 0\). Polynomial equations include terms which are variable powers multiplied by coefficients. Understanding the structure of polynomial equations is key:

• **Degree**: Determined by the highest power of the variable, it indicates the highest number of solutions the equation can have. In \(P(x) = x^3 + 2x - 3\), the degree is 3.
• **Coefficients**: These are the constants multiplying the variables. For example, 2 in front of \(x\) in \(P(x) = x^3 + 2x - 3\) is a coefficient.
• **Roots/Solutions**: These are the values of \(x\) that satisfy the equation. Finding them typically involves factoring, graphing, or using formulas depending on the degree and complexity.

Polynomial equations form the basis of many complex algebraic operations, and solutions can often be found using various algebraic strategies. Becoming comfortable with polynomial equations helps visualize and solve algebra problems efficiently.

Algebraic Operations

Algebraic operations involve basic math processes like addition, subtraction, multiplication, and division, but applied to algebraic expressions instead of just numbers. They are essential to solving problems involving polynomials. Here's how they work in evaluating functions:

• **Addition/Subtraction**: Combine like terms, which are terms with the same variable power. This was seen in simplifying \(-8 - 4 - 3\) to \(-15\) in the solution.
• **Multiplication**: Distribute and multiply constants by terms as needed. For example, \(2(-2) = -4\) used that principle.
• **Exponentiation**: Apply rules of exponents, especially when dealing with negative bases, such as in \((-2)^3\) becoming \(-8\).

These operations, particularly as applied to polynomials, are foundational in algebra. With practice, these operations allow for the manual work required to simplify and evaluate more intricate equations and expressions.