Problem 77 - Algebra 2

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ f(x) \cdot g(x) $$

Step-by-Step Solution

Answer

The result of the operation \(f(x) \cdot g(x)\) is \(3x^3 - 2x^2 + 3x - 2\)

Step 1: Substitute Functions

We start our process by substituting the functions into the required operation. So, \(f(x) \cdot g(x)\) becomes \((3x-2) \cdot (x^2+1)\)

Step 2: Apply Distributive Property

To simplify, use the distributive property, also known as the law of distribution. It states that for all real numbers a, b and c: a \cdot (b + c)= ab + ac. So we can expand \((3x-2) \cdot (x^2+1)\) to \(3x \cdot x^2 - 2 \cdot x^2 + 3x \cdot 1 - 2 \cdot 1 \)

Step 3: Simplify the Equation

Now, simplify the equation by carrying out the operations: \(3x \cdot x^2 - 2 \cdot x^2 + 3x \cdot 1 - 2 \cdot 1\) becomes \(3x^3 - 2x^2 + 3x - 2\)

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to multiply a single term across terms inside a parenthesis. This property simplifies expressions and is crucial for understanding advanced algebraic operations. It states that for any real numbers, if you have an expression like \( a \cdot (b + c) \), you can distribute \( a \) across both \( b \) and \( c \) to get \( ab + ac \).

In the context of function operations, the distributive property facilitates the multiplication of linear expressions with polynomials, as we did in our exercise. To illustrate, the expression \((3x-2) \cdot (x^2+1)\) uses the distributive property to create individual multiplication tasks:

• Multiply \( 3x \) by \( x^2 \), then \( 3x \) by \( 1 \).
• Multiply \(-2 \) by \( x^2 \), then \(-2 \) by \( 1 \).

The result is a polynomial \( 3x^3 - 2x^2 + 3x - 2 \). Each multiplication has been simplified, showing how this property efficiently expands expressions.

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of variables, multiplied by coefficients. A typical polynomial function of one variable, like \( x \), is structured as \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). Here, \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are real number coefficients, and each term's power represents its degree.

In our exercise, \( g(x) = x^2 + 1 \) is a polynomial function where \( x^2 \) represents a quadratic term, and \( 1 \) is the constant term. When we perform operations, like multiplying \((3x-2)\) by this polynomial, we apply the distributive property to expand and simplify the expression into a new polynomial. This process involves carrying out multiplications:

• \( 3x \) interacting with \( x^2 \) creates \( 3x^3 \).
• \( -2 \cdot x^2 \) becomes \(-2x^2 \).
• Terms \( 3x \) and \(-2 \) result in \( 3x - 2 \).

Thus, polynomial functions provide a versatile means of modeling various relationships and patterns in mathematics.

Real Numbers

Real numbers form a foundational set of numbers used in everyday mathematics and more complex theories. This set includes all rational and irrational numbers, encompassing integers, fractions, and numbers with non-repeating, non-terminating decimals.

Real numbers are crucial in function operations, allowing us to handle and express solutions like the polynomial \( 3x^3 - 2x^2 + 3x - 2 \) in the exercise. When manipulating expressions:

• Each coefficient and constant term in polynomial functions originates from the realm of real numbers.
• Real numbers enable operations such as addition, subtraction, multiplication, and division within these expressions.

The properties of real numbers ensure that our solutions are meaningful and consistent, even as we move towards more abstract mathematical concepts. By applying real numbers to function operations, each step, like using the distributive property or simplifying polynomials, is grounded in a solid, rational understanding of mathematics.