Problem 7 - Precalculus: Mathematics for Calculus

Question

Find the domain of the expression. $$4 x^{2}-10 x+3$$

Step-by-Step Solution

Answer

The domain is all real numbers, \((- fty, fty)\).

Step 1: Understand the Type of Expression

The given expression is a quadratic polynomial of the form \(4x^2 - 10x + 3\). Quadratic polynomials are continuous and defined for all real numbers.

Step 2: Identify the Domain of the Polynomial

Since the expression \(4x^2 - 10x + 3\) is a quadratic polynomial, and polynomials are defined for every real number, the domain is all real numbers.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a type of polynomial that includes an expression of degree two. This means it has an exponent of 2 in its highest term. The general form of a quadratic polynomial can be written as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The term \( ax^2 \) is what makes it quadratic.
Quadratic polynomials have a specific shape on a graph called a parabola—a smooth, U-shaped curve. They are fundamental in algebra due to their simplicity and wide range of applications in physics, engineering, and economics.
Quadratic polynomials have properties such as:

• The vertex, which is the highest or lowest point on the graph depending on the direction of the parabola.
• The axis of symmetry, a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• The roots or solutions, which are the values of \( x \) that make the polynomial equal to zero. These are found using the quadratic formula or factoring, where applicable.

Understanding quadratic polynomials requires recognizing their coefficients, structure, and the role each term plays.

Real Numbers

Real numbers include all the numbers you can think of and many more! They encompass whole numbers, fractions, decimals, and even roots like \( \sqrt{2} \). Essentially, real numbers fill the number line without any gaps.
Real numbers are important in determining the domain of functions. When a polynomial function like a quadratic is involved, its domain encompasses all these real numbers.
Real numbers can be categorized into:

• Integers: Includes positive and negative whole numbers, as well as zero.
• Rational Numbers: Numbers that can be expressed as the fraction of two integers.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction, such as \( \pi \) and \( \sqrt{2} \).

Thus, when we state the domain of a polynomial function as all real numbers, we are essentially saying it is defined and continuous across the entire spectrum of numbers you can place on a number line.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable with coefficients. The general form of a polynomial of degree \( n \) is \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are constants and \( n \) is a non-negative integer.
These functions are incredibly versatile and appear often across different branches of science and mathematics because:

• They can be plotted as smooth, continuous curves.
• They encompass a variety of special functions, including linear functions and quadratic functions.
• Each term in a polynomial function has an impact on its growth and behavior as \( x \) changes.

Polynomial functions are defined for all real numbers since they are continuous without any restrictions on \( x \). This universality makes them important for modeling real-world problems and analyzing various situations mathematically.