Problem 47 - Algebra A Combined Function

Question

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7,\) find each function. $$ P(x)+Q(x) $$

Step-by-Step Solution

Answer

The function \( P(x) + Q(x) \) is \( 4x^2 - 3x + 6 \).

Step 1: Understanding the Problem

The exercise asks us to find the sum of two functions, specifically, the function \( P(x) \) and function \( Q(x) \). We are provided with the expressions for each function.

Step 2: Given Function Expressions

We are given the expressions of the functions as follows:\( P(x) = 3x + 3 \)\( Q(x) = 4x^2 - 6x + 3 \)

Step 3: Add the Functions

To find \( P(x) + Q(x) \), we need to add the expressions of both functions together:\[ P(x) + Q(x) = (3x + 3) + (4x^2 - 6x + 3) \]

Step 4: Combine Like Terms

Combine the like terms in the expression:- Combine the \( x^2 \) terms: \( 4x^2 \)- Combine the \( x \) terms: \( 3x - 6x = -3x \)- Combine the constant terms: \( 3 + 3 = 6 \)So, the sum is:\[ P(x) + Q(x) = 4x^2 - 3x + 6 \]

Key Concepts

Polynomial Expressions

A polynomial expression is a mathematical phrase that can consist of constants, variables, and exponents combined using addition, subtraction, and multiplication. They play a crucial role in algebra and help us model various real-world situations. In the context of our exercise, both functions \( P(x) \) and \( Q(x) \) are polynomial expressions given by \( P(x) = 3x + 3 \) and \( Q(x) = 4x^2 - 6x + 3 \).

Here are some key traits of polynomial expressions:

• They contain terms with variables raised to non-negative integer exponents, such as \( x^2 \), \( x \), etc.
• Each term is a product of a constant and a power of a variable, like \( 4x^2 \).
• The terms are combined using addition or subtraction.

This fundamental understanding will allow us to manipulate and combine these expressions in problems such as function addition.

Combining Like Terms

When working with polynomial expressions, combining like terms is a fundamental step. It simplifies expressions and makes calculations more manageable. Like terms are terms that have the same variable raised to the same power. For example, \( 3x \) and \( -6x \) are like terms because they both have the variable \( x \) raised to the first power.

To combine like terms when performing function addition, follow these steps:

• Identify the terms with the same variable and exponent.
• Add or subtract the coefficients of these like terms.
• Repeat the process for each set of like terms in the expression.

In our exercise, after performing the addition, we combine like terms as follows:

• The \( x^2 \) term is \( 4x^2 \)
• The \( x \) terms: \( 3x - 6x = -3x \)
• The constant terms: \( 3 + 3 = 6 \)

This results in a simplified expression: \( 4x^2 - 3x + 6 \).

Algebraic Functions

Algebraic functions are expressions that involve algebraic operations such as addition, subtraction, multiplication, and division on variables. They provide a flexible framework to solve various mathematical and real-world problems.

In our given exercise, we were tasked with finding the sum of two algebraic functions, \( P(x) \) and \( Q(x) \). The emphasis was on combining their given expressions by using simple algebraic operations primarily focusing on addition.

These types of problems frequently require:

• Reading and understanding each polynomial expression that represents a function.
• Performing arithmetic operations following algebraic rules.
• Simplifying the expression by combining like terms.

By practicing these kinds of exercises, students develop a solid grasp of manipulating algebraic functions, which is crucial for more advanced math courses.