Find the Degree, Leading Term, and Leading Coefficient x/9-1
The given problem asks for three specific characteristics of the algebraic expression x/9 - 1:
Degree: This refers to the highest power of the variable (x in this case) in a polynomial expression. The degree indicates the highest exponent.
Leading Term: This is the term in the polynomial that contains the highest power of the variable. It is called the "leading" term because it usually leads the expression when it is written in standard form (i.e., descending powers of x).
Leading Coefficient: This is the numerical factor (coefficient) that multiplies the variable in the leading term. It provides information on the term's magnitude and its influence on the shape and behavior of the graph of the polynomial.
The provided expression needs to be analyzed for these characteristics without actually performing the calculations to find them.
$\frac{x}{9} - 1$
Answer
Solution:
Step 1: Determine the Polynomial Degree
The degree of a polynomial is defined by the highest power of the variable present in the polynomial.
For the term $\frac{x}{9}$, the exponent of $x$ is $1$.
For the constant term $-1$, the exponent of $x$ is $0$ (since $-1$ can be thought of as $-1 \cdot x^0$).
The highest exponent among the terms is $1$, so the degree of the polynomial is $1$.
Step 2: Identify the Leading Term
The leading term of a polynomial is the term with the highest power of the variable.
- In this case, the leading term is $\frac{x}{9}$.
Step 3: Find the Leading Coefficient
The leading coefficient is the coefficient of the term with the highest power of the variable.
- The coefficient of the leading term $\frac{x}{9}$ is $\frac{1}{9}$.
Step 4: Compile the Results
- Polynomial Degree: $1$
- Leading Term: $\frac{x}{9}$
- Leading Coefficient: $\frac{1}{9}$
Knowledge Notes:
To solve for the degree, leading term, and leading coefficient of a polynomial, one must understand the following concepts:
Degree of a Polynomial: The degree is the highest power of the variable in the polynomial. For a single variable polynomial, it is simply the largest exponent the variable has in any term.
Leading Term: This is the term in the polynomial with the highest degree. When a polynomial is written in standard form (terms in descending order of power), the leading term is the first term.
Leading Coefficient: This is the coefficient of the leading term. It is the numerical factor that multiplies the variable in the leading term.
Coefficients: These are the numbers that multiply the variables or powers of variables in polynomial expressions.
Constants: These are terms in the polynomial that do not contain variables. They can be considered as terms with a variable raised to the power of $0$.
Polynomials in Standard Form: Polynomials are often written with the terms in descending order of their degrees. This makes it easier to identify the leading term and coefficient.
When working with polynomials, it's important to remember that every term has a degree, even constants (which have a degree of $0$). The leading term and coefficient play a significant role in the behavior of the polynomial, especially in graphing and in understanding the end behavior of the polynomial's graph.
