Problem 12 - Algebra 2

Question

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(7-x\)

Step-by-Step Solution

Answer

Degree: 1, Leading Coefficient: -1.

Step 1: Identify Terms of the Polynomial

The given expression is \(7 - x\). This expression has two terms: \(7\) and \(-x\). It is written in terms of one variable \(x\).

Step 2: Confirm Polynomial in One Variable

A polynomial in one variable will only have terms with whole number exponents of the variable. The term \(-x\) is equivalent to \(-1x^1\), confirming that both terms are consistent with a polynomial structure.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. Here, \(-x\) has an exponent of 1, and \(+7\) is a constant term with an implied exponent of 0. Thus, the degree is 1.

Step 4: Identify the Leading Coefficient

The leading term in this polynomial is \(-x\) as it has the highest degree. The coefficient of this term is \(-1\), thus the leading coefficient is \(-1\).

Key Concepts

Polynomial Degree

Understanding the **degree of a polynomial** is crucial. It measures the highest power of the variable in the polynomial expression. For example, in the polynomial expression \(7 - x\), the degree is determined by looking for the highest exponent of the variable \(x\).

Here's how you determine it:

• Identify each term in the polynomial that includes the variable.
• The term \(-x\) in our case is actually \(-1x^1\).
• The highest exponent present in the polynomial is 1, making it a first-degree polynomial.

Understanding that a constant value like \(7\) has an implicit degree of 0 is also essential.

Knowing the degree helps predict the polynomial's behavior and its graph's shape.

Leading Coefficient

The **leading coefficient** in a polynomial indicates the coefficient of the term with the highest degree. Once you know the degree of the polynomial, identifying the leading coefficient becomes straightforward.

For the expression \(7 - x\):

• The leading term is \(-x\), since it has the highest degree, which is 1.
• The coefficient of \(-x\) can be expressed as \(-1\) (because \(-x = -1x\)).
• Thus, the leading coefficient of this polynomial is \(-1\).

Leading coefficients are vital as they influence the opening direction of some polynomial graphs and show the influence of the most significant term.

Polynomial Terms

**Polynomial terms** are parts of a polynomial expression that can be added together. Each term consists of a coefficient and a variable raised to an exponent, unless it is a constant term.

In our example \(7 - x\):

• There are two terms: \(7\) and \(-x\).
• \(7\) is the constant term, meaning it has no variable attached, hence an implied degree of 0.
• \(-x\) is another term where the variable \(x\) is raised to the power of 1.

These terms can be combined in different ways to form polynomials, and understanding each component helps in analyzing and solving polynomial expressions effectively.