Problem 51 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The function is \(5x^2 + 25x\).

Step 1: Identify the Task

We need to find the function for the multiplication of two given functions, specifically \(P(x)\) and \(R(x)\).

Step 2: Recall the Function Definitions

Recall that \(P(x)=5x\) and \(R(x)=x+5\). We will use these definitions to replace \(P(x)\) and \(R(x)\) in the multiplication.

Step 3: Substitute and Multiply the Expressions

Substitute \(P(x)=5x\) and \(R(x)=x+5\) into \(P(x) \cdot R(x)\):\[P(x) \cdot R(x) = (5x) \cdot (x+5)\]Now distribute \(5x\) across \(x+5\):\[(5x) \cdot (x+5) = 5x \cdot x + 5x \cdot 5\]Calculate each term: \(5x \cdot x = 5x^2\) and \(5x \cdot 5 = 25x\).

Step 4: Combine the Results

Combine the calculated terms from the previous step:\[5x^2 + 25x\]So, the function for \(P(x) \cdot R(x)\) is \(5x^2 + 25x\).

Key Concepts

Polynomials

Polynomials are mathematical expressions involving a sum of powers of one or more variables, each multiplied by a coefficient. In simple terms, they are like expressions that have variables raised to whole number exponents.
Examples of polynomials include:

• Linear polynomial: a polynomial of degree 1, such as \(x + 5\) or \(3x - 2\).
• Quadratic polynomial: a polynomial of degree 2, such as \(x^2 - 2x + 1\).
• Higher-degree polynomials: such as \(x^3 + x^2 - x + 7\).

Polynomials can have one or several terms and are used to represent a wide range of mathematical situations. A key feature of polynomials is that their exponents are whole numbers and they are classified based on their degree, which is the highest exponent in the expression.

Polynomial Functions

Polynomial functions are functions that are defined by polynomial expressions. The general form of a polynomial function can be expressed as \(a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0\). Here, each \(a_i\) is a coefficient, and \(n\) is the degree of the polynomial.
Features of polynomial functions include:

• Continuity: They are continuous and smooth, with no breaks or holes.
• Domain: The domain of a polynomial function is all real numbers.
• Behavior: The degree dictates end behavior and the number of roots.

Polynomial functions are used to model real-world phenomena and can be combined in various ways, such as addition, subtraction, and importantly—multiplication, as seen with the function multiplication in the given exercise.

Distributive Property

The distributive property is a fundamental rule in algebra that helps us simplify expressions and solve equations. This property states that multiplying a sum by a number is the same as multiplying each addend separately by the number, then adding the results. Formally, it can be expressed as \( a(b + c) = ab + ac \).
When multiplying polynomials, the distributive property is used to ensure that each term in one polynomial is multiplied by each term in the other polynomial. This is how we evaluated the product of \(P(x)\) and \(R(x)\) in the exercise:

• Multiply \(5x\) by \(x\) to get \(5x^2\).
• Multiply \(5x\) by \(5\) to get \(25x\).

By combining these results, we obtain the polynomial function for \(P(x) \cdot R(x) = 5x^2 + 25x\).

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They do not have an equality sign like equations but represent a value that can change based on the variables' values.
In algebraic expressions:

• Variables represent unknown or changeable values, typically shown as letters like \(x\), \(y\), etc.
• Coefficients are numerical factors multiplied by the variables, like the \(5\) in \(5x\).
• Constants are fixed values that do not change, like \(5\) in \(x + 5\).

Algebraic expressions can be simplified through operations like adding like terms or using the distributive property, allowing us to solve problems more efficiently. Understanding how to manipulate these expressions is key to mastering algebra and functions.