Problem 2 - Algebra 2

Question

Determine whether each expression is a polynomial. If is a polynomial, state the degree of the polynomial. $$ \frac{1}{3} x^{3}-9 y $$

Step-by-Step Solution

Answer

The expression is a polynomial with a degree of 3.

Step 1: Identify Terms

The expression \( \frac{1}{3} x^{3}-9y \) consists of two terms: \( \frac{1}{3} x^{3} \) and \(-9y \). Each term is checked for adherence to polynomial conditions, which require whole number powers of variables.

Step 2: Check each Term for Polynomial Conditions

For the term \( \frac{1}{3} x^{3} \), the variable \( x \) is raised to the power of 3, which is a whole number. The term \(-9y \) has the variable \( y \) raised to the power of 1, also a whole number.

Step 3: Verify the Expression as a Polynomial

Since all terms in \( \frac{1}{3} x^{3}-9y \) have variables with non-negative integer powers, the expression is a polynomial.

Step 4: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of a variable present in any term in the expression. The term \( \frac{1}{3} x^{3} \) has the highest power, which is 3.

Key Concepts

Polynomial Degree

Understanding the degree of a polynomial is crucial in identifying its characteristics. The degree is determined by the highest power of the variable in the polynomial. In simpler terms, it's the maximum exponent present in any term of the expression.

A polynomial with a single variable looks like this:

• In the term \( rac{1}{3} x^{3} \), the degree is 3 because the power of \( x \) is 3.
• For a term such as \(-9y\), the degree is 1 since \( y^{1} \) is implied, even if not explicitly written.

The overall degree of a polynomial is the highest degree found across its terms.
So, for the expression \( rac{1}{3} x^{3} - 9y \), the polynomial degree is 3, as this is the highest exponent among the terms.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operational symbols. Unlike equations, they do not have an equals sign. Each expression is made up of terms, which can be a combination of these elements.

A key characteristic of algebraic expressions is that they can have:

• Constant terms, such as the number \(-9 \), which do not change.
• Variable terms, like \( x^{3} \) or \( y \), that involve letters representing numbers.
• Coefficients, such as \( \frac{1}{3} \) and \(-9 \) associated with the variables.

In the example expression \( \frac{1}{3} x^{3} - 9y \), it includes two terms defined by variables \( x \) and \( y \). Understanding how these terms interact is key to manipulating algebraic expressions.

Variables and Coefficients

Variables and coefficients form the building blocks of algebraic expressions. A variable is a symbol that represents an unknown value, often denoted by letters like \( x \) or \( y \). These are placeholders for numbers and can change based on the problem's requirements.

In the expression \( \frac{1}{3} x^{3} - 9y \):

• \( x \) and \( y \) are variables, each representing quantities that can vary.
• The coefficient of a term is the number in front of the variable. For \( \frac{1}{3} x^{3} \), the coefficient is \( \frac{1}{3} \); for \(-9y\), it is \(-9\).

It's essential to note that coefficients can be positive, negative, whole numbers, or fractions. These play an important role in determining the term's value and behavior in an equation or expression.