AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Linear Polynomial Analysis: Classify $6 - \\frac{x}{5}$ and Identify Its Key Components
Mathematics
Grade 9 of Junior High School
Question Content
Answer the questions about the following polynomial: $6 - \\frac{x}{5}$. Fill in the blanks: The expression represents a ______ polynomial with ______ terms. The constant term is ______, the leading term is ______, and the leading coefficient is ______.
Correct Answer
linear; 2; 6; $-\\frac{x}{5}$; $-\\frac{1}{5}$
Detailed Solution Steps
1
Step 1: Classify the polynomial by degree. The highest power of the variable $x$ in $6 - \\frac{x}{5}$ is 1, so it is a linear polynomial.
2
Step 2: Count the number of terms. The polynomial has two distinct terms: $6$ and $-\\frac{x}{5}$, so it has 2 terms.
3
Step 3: Identify the constant term. The constant term is the term without a variable, which is $6$.
4
Step 4: Identify the leading term. The leading term is the term with the highest power of the variable, written in standard form (descending order of exponents). Here, that is $-\\frac{x}{5}$.
5
Step 5: Identify the leading coefficient. The leading coefficient is the numerical coefficient of the leading term, which is $-\\frac{1}{5}$.
Knowledge Points Involved
1
Polynomial Degree Classification
Polynomials are classified by the highest exponent of their variable: a linear polynomial has a highest degree of 1, quadratic has degree 2, cubic has degree 3, etc. This classification describes the shape of the polynomial's graph and its algebraic behavior.
2
Polynomial Terms
A term in a polynomial is a single number, variable, or product of numbers and variables, separated by addition or subtraction signs. For example, in $6 - \\frac{x}{5}$, the terms are $6$ and $-\\frac{x}{5}$.
3
Constant Term
The constant term in a polynomial is the term that does not contain any variables. It remains the same regardless of the value of the variable, and is the y-intercept when the polynomial is graphed as a function.
4
Leading Term and Leading Coefficient
The leading term is the term in a polynomial with the highest power of the variable, written in standard form (descending exponent order). The leading coefficient is the numerical factor of this leading term, which determines the end behavior of the polynomial's graph.
Loading solution...
