Problem 18 - Algebra 2

Question

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ \sqrt{m-5} $$

Step-by-Step Solution

Answer

The expression \( \sqrt{m-5} \) is not a polynomial.

Step 1: Identify the Mathematical Expression

We need to examine the given expression \( \sqrt{m-5} \). This expression involves a square root. Polynomials are algebraic expressions that do not contain variables under radicals, such as square roots.

Step 2: Check for Polynomial Characteristics

Review the characteristics of a polynomial. A polynomial cannot contain variables with negative or non-integer exponents, and it cannot have variables under a radical. Since \( m \) is under a square root in our expression, it does not meet these criteria.

Step 3: Conclude whether Expression is a Polynomial

Given that \( \sqrt{m-5} \) involves a radical with the variable \( m \), it fails to satisfy the definition of a polynomial. Therefore, \( \sqrt{m-5} \) is not a polynomial.

Key Concepts

Degree of a polynomial

The degree of a polynomial is an important concept in algebra that helps us determine the polynomial's characteristics. It tells us about the highest power of the variable in the polynomial. For example, if you have the polynomial expression \( 3x^2 + 4x + 5 \), the degree is 2 because the highest power of \( x \) is 2. This power means that the graph of the polynomial will have a certain number of bends or turns.When looking at polynomials, consider:

• The highest degree of any term with a non-zero coefficient.
• Polynomials may have multiple variables, such as \( 3x^2y + y^2 \). Here, add the exponents of all variables in each term. The degree of \( 3x^2y \) is 3 (2 from \( x^2 \) and 1 from \( y \)).
• If a term is just a number with no variable, its degree is 0.

Understanding the degree helps in comparing polynomials, solving equations, and predicting the shape and behavior of graphs.

Square roots

Square roots are a type of mathematical operation that involve finding a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \). This operation is represented by the symbol \( \sqrt{} \).However, in algebra, square roots become problematic when dealing with polynomials:

• Polynomials are defined as algebraic expressions that do not have variables under a radical (like a square root).
• If an expression contains something like \( \sqrt{m-5} \), it is not considered a polynomial because the variable is under the square root symbol.

Therefore, understanding square roots is crucial to identify whether an expression is a polynomial, as seen in the given exercise.

Algebraic expressions

Algebraic expressions are combinations of variables, constants (numbers), and arithmetic operations such as addition, subtraction, multiplication, and division. They allow us to form equations and inequalities that can be solved or simplified.Here are some key features of algebraic expressions:

• They can include terms like \( 5x \), \( -3 \), or \( 2xy \). Each term is composed of numbers and variables.
• Algebraic expressions can be simplified or factored to make solving easier.
• Expressions differ from equations in that they do not have an equality sign; for example, \( 2x + 3 \) is an expression, while \( 2x + 3 = 7 \) is an equation.

While all polynomials are algebraic expressions, not all algebraic expressions are polynomials. This distinction is important when determining whether an expression like \( \sqrt{m-5} \) fits into the category of a polynomial.