Problem 1 - College Algebra Essentials

Question

In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. $$2 x+3 x^{2}-5$$

Step-by-Step Solution

Answer

Yes, the algebraic expression \(2 x+3 x^{2}-5\) is a polynomial. In standard form, the polynomial is \(3x^2 + 2x - 5\).

Step 1: Identifying the Polynomial

A polynomial is an algebraic expression involving a sum of powers in one or more variables multiplied by coefficients. Here, the expression is indeed a polynomial as it consists of \(2x\), \(3x^2\), and \(-5\), which are powers of the variable \(x\) multiplied by coefficients.

Step 2: Writing the Polynomial in Standard Form

The standard form of a polynomial involves organizing the terms in order of highest degree to the lowest. In our case, the highest degree (order of power of variable x) is 2 and the lowest is 0. Precisely, the standard form should start with \(3x^2\), then \(2x\) and lastly \(-5\).

Key Concepts

Polynomial

Polynomials are foundational elements in algebra that express quantities in a structured way. Imagine them as a collection of numbers often called 'terms', each consisting of a variable raised to an exponent and multiplied by a coefficient. In our example, the algebraic expression is indeed a polynomial because it is composed of such terms. Each term like \(2x\) or \(3x^2\) represents a piece of the full picture, and when combined, they describe a certain quantity completely.

Algebraic Expression

Any mathematical phrase that combines numbers, variables, and operators is known as an algebraic expression. These expressions are the language algebra speaks to convey mathematical ideas. In the provided exercise with the expression \(2x + 3x^2 - 5\), we see a clear example of how algebraic expressions are built. They can be as simple as a single number or as complex as a long combination of terms.

Coefficient

Moving into the more detailed aspects, the coefficient is the number that multiplies a variable in an algebraic expression. It provides a measure of how much the variable term will 'weigh'. For \(2x\), the coefficient is 2, which means we have twice the amount of whatever \(x\) represents. Consequently, if \(x\) changes, the effect on the overall expression is amplified by the coefficient's value.

Variable

A variable acts as a placeholder for values we might not know yet or that can change. In algebra, it's usually represented by a letter such as \(x\), \(y\), or \(z\). In our expression, \(x\) is the variable and it takes on various powers. Variables allow algebraic expressions to be versatile and applicable to many different situations, making them indispensable in mathematics.

Degree of a Polynomial

Understanding the degree of a polynomial is crucial. It is the highest power of the variable in the polynomial. It tells us a lot about the polynomial's characteristics and behavior. For example, a first-degree polynomial like \(2x\) is a straight line when graphed, whereas a second-degree polynomial like \(3x^2\) forms a parabola. In our exercise, the highest degree term is \(3x^2\), making the polynomial a second-degree polynomial. This is significant because the degree will dictate the standard form order, starting with the highest degree and descending to the lowest.