Problem 48 - 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. $$ Q(x)-R(x) $$

Step-by-Step Solution

Answer

\(-x^2 - 6x + 10\)

Step 1: Write Down the Functions

Start by writing down the expressions for both functions involved: \[ Q(x) = 4x^2 - 6x + 3 \] \[ R(x) = 5x^2 - 7 \]

Step 2: Set Up the Subtraction

We need to calculate \(Q(x) - R(x)\). Set up the expression by putting the functions into a single expression:\[ Q(x) - R(x) = (4x^2 - 6x + 3) - (5x^2 - 7) \]

Step 3: Distribute the Negative Sign

Distribute the negative sign through the second function, \(R(x)\), making sure to change the sign of every term:\[ (4x^2 - 6x + 3) - 5x^2 + 7 \]

Step 4: Combine Like Terms

Combine the like terms from the expanded expressions: \[ (4x^2 - 5x^2) + (-6x) + (3 + 7) \] Simplify the expressions:\[ -x^2 - 6x + 10 \]

Step 5: Final Expression

The resulting expression after combining like terms is the simplified version:\[ Q(x) - R(x) = -x^2 - 6x + 10 \]

Key Concepts

Function Subtraction

Function subtraction is a fundamental concept in algebraic functions that allows us to determine the difference between two functions. To perform function subtraction, follow these straightforward steps:

• Write down each function separately. For our exercise, the functions are given as: \(Q(x) = 4x^2 - 6x + 3\) and \(R(x) = 5x^2 - 7\).
• Set up the subtraction by aligning the functions in one combined expression, ensuring each term is considered. Here, we have \((4x^2 - 6x + 3) - (5x^2 - 7)\).
• Apply the subtraction by distributing the negative sign through the second function,\(R(x)\), which effectively changes the sign of each term in \(R(x)\). This gives us \((4x^2 - 6x + 3) - 5x^2 + 7\).
• Once the sign change is applied, proceed to simplify the resulting expression.

By understanding these steps, you'll become skilled at handling subtraction between any two algebraic functions with confidence.

Polynomial Functions

Polynomial functions are a common type of algebraic function composed of terms that are non-negative integer powers of a variable, typically represented as \(x\). For example, \(4x^2\), \(-6x\), and \(3\) are each terms of the polynomial functions in our exercise.

Polynomials are important because they provide the structural foundation for various algebraic operations, including addition, subtraction, and multiplication. Each polynomial can vary in degree, which is the highest power of the variable present. In our case:

• \(Q(x) = 4x^2 - 6x + 3\) is a polynomial of degree 2 (since \(x^2\) is the highest power).
• \(R(x) = 5x^2 - 7\) is also a polynomial of degree 2.

Using polynomials in function operations like subtraction helps you practice managing expressions involving multiple terms of different powers, reinforcing the core strategies of algebraic manipulation.

Like Terms

Combining like terms is an essential practice in simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power. They can be added or subtracted easily because they share these identical characteristics.

In the exercise's context:

• \(4x^2\) and \(-5x^2\) are like terms because both are multiplied by \(x^2\).
• \(-6x\) has no like term in the expression \(R(x)\).
• \(3\) and \(7\) are constants, and they are also considered like terms because they do not interface with the variable \(x\).

The process of combining like terms helps in simplifying the overall expression. As seen, \((4x^2 - 5x^2) + (-6x) + (3 + 7)\) simplifies to \(-x^2 - 6x + 10\).

Mastering the art of identifying and combining like terms is crucial for tackling more complex algebraic expressions efficiently.