Problem 16 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The expression \(x^3 - 9\) is a polynomial with a degree of 3.

Step 1: Identify the Terms

The given expression is \(x^3 - 9\). First, identify the individual terms in the expression. In this case, the two terms are \(x^3\) and \(-9\).

Step 2: Check Polynomial Criteria

Ensure that each term in the expression meets the criteria for being part of a polynomial. A polynomial is composed of terms that have non-negative integer exponents or constants. Here, the exponents are 3 (from \(x^3\)) and 0 (from the constant \(-9\)), which are non-negative integers.

Step 3: Conclusion about Being a Polynomial

Since all terms have non-negative integer exponents, the expression \(x^3 - 9\) is indeed a polynomial.

Step 4: Identify the Degree

The degree of a polynomial is determined by the highest exponent of the variable. In \(x^3 - 9\), the highest exponent is 3, thus the degree of the polynomial is 3.

Key Concepts

Polynomial Degree

The degree of a polynomial is a key concept in understanding polynomials. It is defined as the highest power of the variable in the polynomial expression. For instance, in the expression \(x^3 - 9\), the highest power of the variable \(x\) is 3, which means the polynomial degree is 3.
This concept is crucial because the degree can tell us about the number of roots a polynomial can have and its general shape when graphed. For example:

• A polynomial of degree 3 can have up to 3 roots.
• The curve of the graph will generally have 2 turning points.

Understanding the degree helps in predicting the behavior of polynomials in mathematical functions.

Non-negative Integer Exponents

To determine if an expression is a polynomial, one must check the exponents of each term. A polynomial only includes terms whose exponents are non-negative integers. A non-negative integer is an integer that is zero or positive.
For example, in the expression \(x^3 - 9\), the term \(x^3\) has an exponent of 3, which is a non-negative integer, and the term \(-9\) has an implicit exponent of 0 on its variable (since \(-9\) can be seen as \(-9x^0\)).
Why is this important? In mathematical operations, if you encounter a fraction or a negative exponent, that term cannot be part of a polynomial. Here are some key points:

• Exponents should not be fractions or negative.
• Only integer values that are 0 or higher are allowed.

Following these rules helps correctly identify polynomials.

Constant Terms

In a polynomial, a constant term is a part of the expression that does not contain any variables; it can be considered as having a variable with an exponent of zero. For example, in the expression \(x^3 - 9\), the number \(-9\) is a constant term.
Constant terms are important because they do not change the degree of the polynomial but can influence the behavior of the polynomial when graphed. In many cases:

• Constant terms determine where the polynomial graph intersects the y-axis.
• Even without variables, constant terms contribute to the overall polynomial function.

Knowing how to identify and interpret constant terms is a crucial part of working with polynomial expressions.