Problem 7 - 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

The expression is not a polynomial, so it has no degree.

Step 1: Identify the expression type

The given expression is \( \sqrt{x^{2}+3x+12} \). To determine if it is a polynomial, note that a polynomial is a sum of monomials, each of which is a product of a constant and a non-negative integer power of a variable, without radicals or fractions in exponents. The given expression is a square root of a polynomial.

Step 2: Determine if the expression is a polynomial

Since the expression involves a square root, it is not a polynomial. Polynomials cannot have variables under a radical sign.

Step 3: Conclude on the degree

As the expression is not a polynomial, it does not have a degree. Degrees only apply to polynomial expressions.

Key Concepts

Monomials

Monomials are the simplest building blocks of polynomials. A monomial is a single term that is a product of numbers and variables, where the variables can only have non-negative integer exponents.
For example, examples of monomials include expressions like \(5x^3\), \(-7xy\), and \(3\). Notice that none have variables in the denominator or under a radical sign.

• Each monomial consists of a coefficient (which can be any real number) and a variable with an exponent (like \(x^2\)).
• Monomials cannot have variables with negative or fractional exponents.

They are vital in forming polynomials, as polynomials themselves are simply a sum of one or more monomials. This simple structure makes it easy to perform operations on polynomials and understand their properties.

Degree of a Polynomial

The degree of a polynomial is a key characteristic that helps to understand and describe it. The degree is the highest power of the variable in the polynomial expression.
For instance, consider the polynomial \(2x^4 + 3x^2 + 7\). Here, the term \(2x^4\) gives the polynomial its degree, which is 4, as it is the largest exponent.

• To find the degree, identify the term with the maximum exponent on its variable.
• Only consider the variables, not their coefficients, when determining the degree.

Knowing the degree of a polynomial helps to predict its behavior and the shape of its graph.
In polynomial equations, the degree indicates the number of potential roots, though some may be complex or repeated.

Radicals in Polynomials

Radicals can significantly affect whether an expression qualifies as a polynomial. In mathematics, a radical expression is one containing a root symbol, such as a square root or cube root.
In the context of polynomials, radicals present a limitation. A true polynomial cannot include any radicals involving the variables.

• An expression like \(\sqrt{x^2+3x+12}\) is not a polynomial because it contains a radical encompassing the variable \(x\).
• Polynomials must be free of radicals and should only involve whole number exponents.

Understanding the restriction of radicals in polynomials helps students correctly identify polynomial expressions and avoid common mistakes. This knowledge ensures proper classification and manipulation of expressions in algebra.