Problem 5 - College Algebra

Question

$$\text {find the degree of the polynomial.}$$ $$3 x^{2}-5 x+4$$

Step-by-Step Solution

Answer

Therefore, the degree of the given polynomial \(3x^2 - 5x + 4\) is 2.

Step 1: Identify the Polynomial

The given polynomial is \(3x^2 - 5x + 4\).

Step 2: Identify the Powers of x

In the given polynomial, the powers of 'x' are 2 and 1.

Step 3: Identify the Highest Power

Among the powers 2 and 1, the highest is 2

Key Concepts

Understanding Algebra and Polynomials

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols; it is a unifying thread of almost all of mathematics. In algebra, we often deal with expressions known as polynomials. A polynomial is a mathematical expression made up of variables (like 'x') and coefficients (like the 3 in '3x'), where the variables have whole-number exponents (such as the 2 in 'x^2') and the coefficients are real numbers.

When we work with polynomials, we deal with operations like addition, subtraction, and multiplication. In the exercise, we're given the polynomial '3x^2 - 5x + 4', where each term is made up of a variable raised to a power (an exponent) and a coefficient. Understanding polynomials is essential in algebra as they are used to model a wide range of phenomena and solve various equations.

Powers of Variables in Polynomials

In algebra, the 'power' of a variable represents the exponent to which the variable is raised. It signifies how many times the variable is multiplied by itself. For instance, in the expression 'x^3', 'x' is the variable, and '3' is the power. This tells us that 'x' is multiplied by itself three times: 'x * x * x'.

The term with the highest power determines the degree of the polynomial. Each term in a polynomial like '3x^2' has a variable (in this case, 'x') and a power (here, '2'). It is essential to be comfortable with powers of variables, as they play a crucial role in algebra, especially when simplifying expressions, finding derivatives in calculus, or in solving polynomial equations.

Identifying the Highest Power in a Polynomial

The degree of a polynomial is determined by the term with the highest power of the variable. For example, in the polynomial '3x^2 - 5x + 4', we can see that the term '3x^2' has the variable 'x' raised to the power of 2. The next term '5x' has the power of 1, and the last number '4' is a constant with an implicit power of 0, as any number to the power of 0 is 1 (except for 0 itself).

To identify the degree of the polynomial, we search for the term with the largest exponent. In this case, 'x^2' has the largest exponent, which means the polynomial '3x^2 - 5x + 4' is a second-degree polynomial. Recognizing the highest power quickly allows us to categorize and work with polynomials more effectively in algebra, be it in solving equations or graphing polynomial functions.