Problem 52 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The product \(P(x) \cdot Q(x)\) is \(5x^3 - 10x\).

Step 1: Understand the given functions

We are provided with three functions: \(R(x)=x+5\), \(Q(x)=x^2-2\), and \(P(x)=5x\). For this exercise, we need to focus on the functions \(P(x)\) and \(Q(x)\).

Step 2: Substitute the given functions

We need to calculate the product \(P(x) \cdot Q(x)\). First, write each function: \(P(x)=5x\) and \(Q(x)=x^2-2\).

Step 3: Perform the multiplication

To find \(P(x) \cdot Q(x)\), multiply the expressions: \[P(x) \cdot Q(x) = (5x) \cdot (x^2-2)\].Apply the distributive property: \[5x \cdot x^2 + 5x \cdot (-2)\].

Step 4: Simplify the expression

Calculate each term obtained in the multiplication:1. \(5x \cdot x^2 = 5x^3\)2. \(5x \cdot (-2) = -10x\)Combine them to get: \[P(x) \cdot Q(x) = 5x^3 - 10x\].

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, essential for multiplying expressions efficiently. This property allows us to multiply a single term with each term in a parenthesis separately, and then add or subtract the results. When dealing with polynomial expressions, grasping the distributive property enables us to break down complex problems into simpler parts.

For instance, when we multiply the function representations of polynomials, like in our exercise, we used the distributive property to tackle the product of two polynomials:

• Consider the expression \((5x) \cdot (x^2 - 2)\).
• This is broken down using the distributive property: \(5x \cdot x^2 + 5x \cdot (-2)\).
• Each part (\(5x \cdot x^2\) and \(5x \cdot (-2)\)) is computed separately.

Understanding this concept is crucial for anyone learning algebra because it simplifies expression manipulation and helps solve equations involving more complex algebraic structures.

By mastering the distributive property, students can handle larger expressions confidently and accurately.

Functions in Algebra

In algebra, functions are like mathematical machines where you input a number and get another number after applying a certain rule. Functions help us understand and predict behavior of mathematical models, especially when representing relationships between different quantities.

In the given exercise, we see polynomial functions, which are widely used in algebra. Each function, such as \(R(x) = x + 5\), \(Q(x) = x^2 - 2\), and \(P(x) = 5x\), is composed of algebraic expressions indicating specific operations on the variable \(x\).

Understanding functions in algebra involves:

• Identifying function types based on their expressions, such as linear, quadratic, and exponential functions.
• Using function notation effectively, e.g., \(f(x)\), to describe inputs and outputs.
• Evaluating functions by substituting values into expressions.
• Working with operations on functions, such as addition, subtraction, multiplication, and division.

The key is to become familiar with how these operations alter the graphs and outputs of the functions. Recognizing the purpose of each function type enables you to choose appropriate strategies for solving problems involving these mathematical models.

Simplifying Expressions

Simplifying algebraic expressions is the process of reducing them into their simplest form. This technique helps make complex calculations easier and more manageable. Simplified expressions are clearer and more concise, often offering direct insights into the behavior of the algebraic formula.

In the context of polynomial expressions, such as those in our exercise, simplifying involves combining like terms and ensuring the expression is as compact as possible. Let's break down the steps in our exercise:

• First, we performed multiplication using the distributive property: \[5x \cdot x^2 = 5x^3\] and \[5x \cdot (-2) = -10x\].
• Next, combining these results gave the expression: \[5x^3 - 10x\].
• This expression is simplified by ensuring all like terms are combined, which in this case, it's already in its simplest form.

Understanding how to simplify equations is essential for solving algebraic problems efficiently. It helps clarify mathematical reasoning and enables students to focus on solving equations without unnecessary complexity. Students should practice identifying terms to combine and simplify expressions as much as possible to develop proficiency in this fundamental skill.