Problem 1 - Algebra and Trigonometry

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 given algebraic expression is a polynomial and its standard form is \(3x^2 + 2x - 5\).

Step 1: Identity of expression

Identify if the given expression is a polynomial. A polynomial is an expression of the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_i's\) are constants and the exponents are non-negative integers. Here, the expression \(2x + 3x^2 - 5\) can be identified as a polynomial because it follows the aforementioned format, with non-negative integer exponents and constant coefficients.

Step 2: Arrange in standard form

Arrange the polynomial in standard form by placing the terms in decreasing order of their degrees. In this case, rearrange \(2x + 3x^2 -5\) to \(3x^2 + 2x - 5\) which is the standard form.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs like '+' and '-'. They are essential in algebra as they form the basis of equations and inequalities. An algebraic expression might look like this: \(2x + 3x^2 - 5\). In this expression, '2x', '3x^2', and '-5' are called terms. Each term has a coefficient (the number in front) and a variable part (like \(x\) or \(x^2\)).

Algebraic expressions can become more complex with additional operations such as multiplication and division. Furthermore, they form the foundation for higher mathematical concepts. By manipulating algebraic expressions, we can solve equations and model real-world situations.
Understanding algebraic expressions is the first step toward recognizing more complex structures like polynomials.

Standard Form of Polynomials

A polynomial is a specific type of algebraic expression with constants and variables raised to non-negative integer exponents. To communicate effectively, especially in mathematics, having a uniform way to present polynomials is crucial.

The standard form of a polynomial is achieved by arranging its terms in decreasing order of their exponents, from the highest degree to the lowest, and writing their coefficients accordingly. For example, the expression \(2x + 3x^2 - 5\) is rearranged into \(3x^2 + 2x - 5\), which is its standard form.

• Start by identifying the degree of each term (based on the exponent).
• Arrange the terms with the highest degree first, proceeding to the lowest.
• Ensure each term's sign (positive or negative) is retained during the arrangement.

Using the standard form gives clarity and uniformity, making it easier for others to follow your work and understand your calculations.

Non-negative Integer Exponents

Exponents are a fundamental component of polynomials and algebraic expressions. They indicate how many times a number (the base) is multiplied by itself. However, in the context of polynomials, we limit the discussion to non-negative integer exponents.

A non-negative integer exponent is an integer greater than or equal to zero. The exponents of polynomials can be zero or any positive integer, but never negative or fractional. This restriction ensures that every polynomial is a valid algebraic expression with terms that can be easily managed.

• Zero exponent: Any number raised to the power of zero is one (e.g., \(x^0 = 1\)).
• Positive integer exponents: Indicate repeated multiplication of the base (e.g., \(x^2 = x \times x\)).

Working with non-negative exponents ensures that operations with polynomials remain straightforward, and the expressions remain meaningful within the context of algebra.