Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The _____ coefficient of \(x^{2}-3 x+2\) is 1

Step-by-Step Solution

Answer

The leading coefficient of \(x^{2} - 3x + 2\) is 1.

Step 1: Identify the Coefficients in the Polynomial

The given polynomial is \(x^{2} - 3x + 2\). It is a quadratic polynomial, which can be expressed in the general form \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are coefficients. In this form, \(a\) corresponds to the coefficient of \(x^{2}\), \(b\) to the coefficient of \(x\), and \(c\) is the constant term.

Step 2: Determine the Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. Since \(x^{2}\) is the highest degree term in \(x^{2} - 3x + 2\), the coefficient of \(x^{2}\) is the leading coefficient.

Step 3: Substitute and Fill in the Blank

The leading coefficient of \(x^{2}\) in the polynomial \(x^{2} - 3x + 2\) is 1. Substitute this value in the blank given in the exercise sentence.

Key Concepts

Quadratic Polynomials

Quadratic polynomials are a commonly studied form of polynomial, particularly in algebra. You may recognize a quadratic polynomial by its characteristic feature of having three terms and a specific degree of 2. In its standard expression, a quadratic polynomial is typically written as \(ax^2 + bx + c\), where:

• \(a\), \(b\), and \(c\) are constants referred to as coefficients.
• \(x\) is the variable, and \(x^2\) is the term with the highest degree, contributing to its name as a quadratic.

It's essential to note that the degree of a polynomial corresponds to the highest power of the variable present. The quadratic polynomial focuses on second-degree expressions. This form is especially useful in various mathematical and real-world applications, including solving equations pertaining to motion and determining areas under parabolic curves.

Leading Coefficient

Identifying the leading coefficient is a crucial part of understanding polynomials. The leading coefficient is the coefficient of the term with the highest power. When writing a polynomial in standard form, such as \(ax^2 + bx + c\), \(a\) is considered the leading coefficient.

• It plays a vital role in determining the shape and orientation of the parabola represented by the quadratic polynomial.
• If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards.
• In the polynomial \(x^2 - 3x + 2\), the leading coefficient is 1, indicating an upward opening parabola.

Understanding the leading coefficient helps in graphing and analyzing the behavior of quadratic functions.

Algebraic Expressions

Algebraic expressions like quadratic polynomials are foundational to algebra and involve combinations of variables and constants using arithmetic operations. Understanding these expressions is essential, particularly since they:

• Form the basis for equations that we solve to find unknown values.
• Involve operations such as addition, subtraction, multiplication, and division.

In a quadratic expression like \(x^2 - 3x + 2\), the combination of terms represents a relationship or rule that can model a variety of real-world phenomena. These expressions can be manipulated through factorization, expansion, and simplification, making them versatile tools in mathematics. Recognizing and working with coefficients and terms in algebraic expressions is key to mastering algebra.