Problem 15 - Algebra and Trigonometry

Question

Determine whether the expression is a polynomial. If it is, state its degree. \(\frac{1}{3} x^{3}+\sqrt{5} x-\frac{1}{4}\)

Step-by-Step Solution

Answer

Yes, it is a polynomial with degree 3.

Step 1: Identify the Types of Terms

The given expression is \( \frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4} \). We observe that it contains different terms. The first term is \( \frac{1}{3} x^{3} \), the second term is \( \sqrt{5} x \), and the third term is a constant \( -\frac{1}{4} \). Each term is either a constant or involves a variable raised to a whole number exponent.

Step 2: Check for Polynomial Criteria

For an expression to be a polynomial, all variable exponents must be non-negative integers, and coefficients can be any real number (including fractions or irrational numbers). The term \( \frac{1}{3} x^3 \) is a polynomial term because it has a non-negative integer exponent (3) and a real coefficient (\( \frac{1}{3} \)). The term \( \sqrt{5} x \) also meets the polynomial criteria with an exponent of 1 and a real coefficient (\( \sqrt{5} \)). The constant \( -\frac{1}{4} \) is also a valid polynomial term, as constants are considered polynomials of degree 0.

Step 3: Determine if Expression is a Polynomial

Since all terms in the expression \( \frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4} \) meet the criteria for being polynomial terms (each variable has a non-negative integer exponent), the entire expression is indeed a polynomial.

Step 4: Identify the Degree of the Polynomial

The degree of a polynomial is determined by the term with the highest power of the variable. In \( \frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4} \), the term \( \frac{1}{3} x^3 \) has the highest exponent, which is 3. Therefore, the degree of this polynomial is 3.

Key Concepts

Polynomial Degree

Understanding the degree of a polynomial is essential in determining the behavior and form of the polynomial. The degree is defined as the highest power of the variable within the polynomial expression.

• For example, in the polynomial expression \(\frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4}\), the degree is 3 because the term \(\frac{1}{3} x^3\) has the highest exponent, which is 3.
• An easy way to identify the degree is to look for the largest exponent of the variable.
• Note that constant terms have a degree of 0, since they do not contain a variable.

This measure is crucial as it helps in classifying polynomials for different mathematical analyses and operations.

Polynomial Terms

A polynomial is a mathematical expression made up of terms added or subtracted together. Each term consists of a coefficient, a variable, and an exponent.

• In the expression \(\frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4}\), there are three terms.
• The first term is \(\frac{1}{3} x^{3}\), the second term is \(\sqrt{5} x\), and the third term is the constant \(-\frac{1}{4}\).
• Each term can be classified based on the exponent and the type of coefficient it has.

Polynomials can have several terms, and understanding each one's role and form is essential in simplifying or solving polynomial equations.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In the context of polynomials, exponents dictate the degree of each term.

• In the expression \(\frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4}\), the first term has an exponent of 3, which is the highest in the polynomial.
• The term \(\sqrt{5} x\) has an exponent of 1, and constant terms like \(-\frac{1}{4}\) have an implicit exponent of 0.

Exponents must be non-negative integers in a polynomial. They determine not just the degree, but also impact the graph shape and the number of solutions that an equation might have.

Real Coefficients

A coefficient is the numerical part of a term, and real coefficients can be any number from the set of real numbers.

• In our example \(\frac{1}{3} x^{3} + \sqrt{5} x - \frac{1}{4}\), the coefficients are \(\frac{1}{3}\), \(\sqrt{5}\), and \(-\frac{1}{4}\).
• These coefficients can include fractions or irrational numbers and are not limited to integers.

Real coefficients provide flexibility in different contexts, making polynomials applicable in various practical scenarios. They affect amplitude and the vertical stretch or compression of a polynomial's graph.