Problem 13 - Algebra and Trigonometry

Question

Determine whether the expression is a polynomial. If it is, state its degree. \(\sqrt{x^{2}+3 x+12}\)

Step-by-Step Solution

Answer

No, it's not a polynomial.

Step 1: Understand Polynomials

A polynomial is an algebraic expression that consists of variables, coefficients, and the operations of addition, subtraction, and multiplication, with non-negative integer exponents. For example, expressions like \(x^2 + 3x + 12\) are polynomials.

Step 2: Identify the Given Expression

The given expression is \(\sqrt{x^2 + 3x + 12}\). You need to determine whether it can be expressed as a polynomial.

Step 3: Analyze the Expression

The expression under the square root is \(x^2 + 3x + 12\), which on its own is a polynomial of degree 2. However, the entire expression includes the square root of this polynomial.

Step 4: Determine if the Expression is a Polynomial

Polynomials cannot include square roots of variable terms. Therefore, \(\sqrt{x^2 + 3x + 12}\) is not a polynomial because it includes a square root that affects the variable.

Key Concepts

Polynomial Degree

When we talk about the *degree of a polynomial*, we refer to the highest power of the variable in the polynomial expression. For example, in the polynomial expression \(x^2 + 3x + 12\), the highest power of \(x\) is 2, thus making the degree 2.
Degrees help us understand the behavior of polynomials, especially as the variable \(x\) grows large.
It's crucial to note that the degree of a polynomial is always a non-negative integer. This is because polynomials only involve whole number exponents for their variables.
When identifying the degree in more complex expressions, always focus on the term with the highest power; if there's a term like \(x^5\), the degree is 5. Remember that the entire expression must fit the criteria for being a polynomial to count the degree.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They form the core building blocks in algebra.
An algebraic expression can be as simple as \(x + 2\) or as complex as \(3x^3 + 5x^2 - 7x + 10\).
Importantly, polynomials are a special type of algebraic expression. To qualify as a polynomial, the expression must use only whole number exponents.
Thus, an implicit requirement in maintaining the term **polynomial** is avoiding operations that introduce non-integer or negative exponents, such as division by a variable or taking roots of variables, like square roots or cube roots.

• Conforms with addition, subtraction, multiplication, and whole number exponents.
• Polynomials are simplified algebraic expressions.

Non-negative Integer Exponents

Non-negative integer exponents are simply whole-number exponents that are zero or positive. They are a key feature of polynomial expressions.
For instance, consider the expression \(x^3 + 4x^2 - 7\). Here, the exponents are 3 and 2, which are clearly non-negative integers.
While handling polynomials, ensure each term in the expression consists of non-negative integer exponents. This is vital because a polynomial cannot have variables raised to negative, fractional, or decimal powers.

• Non-negative integers include 0, 1, 2, 3, and so forth.
• They assure the expression remains a polynomial when used as exponents.
• Expressions like \(x^{-1}\) or \(x^{1/2}\) do not satisfy this condition and hence do not form polynomials.

Understanding this concept is crucial for correctly identifying and classifying polynomial expressions.