Problem 1 - Elementary and Intermediate Algebra

Question

The ____ coefficient of \(3 x^{2}-x-12\) is 3

Step-by-Step Solution

Answer

The leading coefficient of \(3x^{2} - x - 12\) is 3.

Step 1: Understand the structure of a polynomial

A polynomial is expressed in the form of \(ax^n + bx^{n-1} + \, ... \, + zx^0\), where each term consists of a coefficient (a number) and a variable \(x\) with a non-negative integer exponent.

Step 2: Identify the given polynomial

The polynomial given is \(3x^{2} - x - 12\). It is a quadratic polynomial because the highest power of \(x\) is 2.

Step 3: Locate the target term from the polynomial

The problem asks for the __ coefficient of \(3x^{2} - x - 12\), and we are provided with both the polynomial and its leading coefficient.

Step 4: Find the leading coefficient

In a polynomial, the leading coefficient is the coefficient of the term with the highest power of \(x\). For \(3x^{2} - x - 12\), the term with the highest power of \(x\) is \(3x^{2}\), whose coefficient is 3.

Key Concepts

Quadratic Polynomial

When dealing with polynomials, understanding their type and degree is crucial. A quadratic polynomial is specifically a polynomial of degree 2. This means that the highest power of the variable, usually denoted as \(x\), is 2. Such polynomials have the general form:

• \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).

This structure is what differentiates quadratic polynomials from linear (Math degree of 1) and cubic (degree of 3) polynomials. Quadratic polynomials can be found in many contexts, especially in geometric problems involving areas or trajectories.
In our specific polynomial, \(3x^2 - x - 12\), it's called quadratic because the highest exponent of \(x\) is 2. The structure clearly follows the quadratic form with \(a = 3\), \(b = -1\), and \(c = -12\).

Leading Coefficient

In any polynomial, the leading coefficient is very important. It is the coefficient of the term with the highest degree.
This term not only dictates how the polynomial behaves as \(x\) becomes very large or very small but also affects the shape of its graph.
For the polynomial \(3x^2 - x - 12\), the leading term is \(3x^2\) because 2 is the highest exponent. Thus, the leading coefficient is the number 3.

• This number reveals the steepness of the parabola in a quadratic polynomial, as well as the direction it opens for parabolas.
• If the leading coefficient is positive, as in our polynomial, the parabola opens upward.
• If it were negative, the parabola would open downward.

Polynomial Structure

Understanding the structure of a polynomial helps in identifying its properties and solving related problems.
A polynomial essentially consists of terms, each with a coefficient and a certain power of the variable \(x\). Here is a quick breakdown:

• The coefficients are constant numbers multiplying each term.
• The exponents on \(x\) in each term show the degree of that term.
• The highest exponent gives the degree of the entire polynomial.

The standard form of a polynomial arranges its terms in descending order of the exponents.
For \(3x^2 - x - 12\), we have:

• \(3x^2\) is the leading term with a coefficient of 3.
• \(-x\) (or \(-1x\)) is the middle term.
• The constant term is \(-12\), with no variable attached.

By understanding each part of a polynomial, you can readily solve for roots, determine behavior, and graph these algebraic expressions.