Problem 1 - College Algebra

Question

In Exercises, 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 given algebraic expression is a polynomial. The standard form is \(3x^2 + 2x - 5\).

Step 1: Identify the expression

Identify the given algebraic expression. The expression here is \(2x + 3x^2 - 5\).

Step 2: Check if it is a polynomial

Check whether the given expression is a polynomial. An expression is a polynomial if it is made up of terms that only have nonnegative exponents, which is the case here.

Step 3: Write in standard form

If the expression is identified as a polynomial, the next thing is to write it in standard form. The standard form of a polynomial is written by rearranging from the highest power to the lowest power of the variable. Rearrange the terms of the polynomial so that it's in the standard form.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations such as addition, subtraction, multiplication, and division. It represents a mathematical phrase that can express a value or an equation. Think of algebraic expressions as a way to describe real-world situations with symbols. For instance, the expression \(2x + 3x^2 - 5\) includes:

• Variables: These are letters like \(x\) that represent unknown values.
• Coefficients: These are numbers placed in front of variables, like 2 and 3 in the expression that multiply the variables.
• Constants: These are fixed numbers without variables attached, like -5 in the expression.

Algebraic expressions can vary in complexity from simple to intricate, depending on the number of terms and operations involved. The goal is often to simplify or manipulate these expressions for various purposes, such as solving equations or modeling real-life scenarios. The key to understanding algebraic expressions is to recognize the components and how they interact within the expression.

Standard Form of Polynomial

The standard form of a polynomial is a method of organizing the expression by arranging its terms in descending order of the power of the variable. This structured way of presenting the polynomial helps in various mathematical processes such as addition, subtraction, and finding roots efficiently. Consider our expression: \(2x + 3x^2 - 5\). To convert this into its standard form, observe the powers of \(x\):

• 3x^2 is the term with the highest power (2),
• 2x has a power of 1,
• -5 is a constant term with a power of 0.

So, the standard form will be \(3x^2 + 2x - 5\), where terms are arranged from the highest degree to the lowest. This format is particularly useful for performing polynomial operations and makes it easier to identify leading coefficients and degrees.

Polynomial Terms

Polynomial terms are the building blocks of a polynomial expression, each comprising a coefficient, a variable, and an exponent. In the polynomial expression \(3x^2 + 2x - 5\), let's break down the terms:

• 3x^2: This is a term where 3 is the coefficient, \(x\) is the variable, and 2 is the exponent, denoting \(x\) raised to the power of 2.
• 2x: Here, 2 is the coefficient, and the exponent of \(x\) is 1, which often isn't written because it's understood.
• -5: This is a constant term, with no variable, thus having an exponent of 0.

Each term in a polynomial is either connected by plus (+) or minus (-) signs, influencing how they combine various mathematical values. Understanding the structure of polynomial terms is crucial for identifying the degree of the polynomial, factoring, and simplifying expressions effectively. By mastering the concept of polynomial terms, one can confidently tackle more complex algebraic challenges.