AP Precalculus Practice Test: Unit 1 Question #9 Degree of a Polynomial and Leading Coefficient

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

**AP Precalculus Practice Test: Unit 1, Question #9: Degree of a Polynomial and Leading Coefficient**

In this question, you are asked to identify the **degree** of a polynomial and the **leading coefficient**. These are important concepts in understanding the behavior of polynomial functions.

### 1. **Degree of a Polynomial**:
The **degree** of a polynomial is the highest power of the variable \( x \) in the polynomial expression. It indicates the polynomial's highest exponent and helps determine its general shape and end behavior.

#### How to Find the Degree:
- Identify the term with the highest power of \( x \).
- The degree of the polynomial is the exponent of this term.

**Example**: For the polynomial \( f(x) = 4x^5 - 3x^3 + 2x^2 - 7 \), the highest power of \( x \) is \( x^5 \), so the degree of the polynomial is **5**.

### 2. **Leading Coefficient**:
The **leading coefficient** is the coefficient of the term with the highest degree in the polynomial. It tells you how steep or flat the graph of the polynomial will be at the ends and affects the direction in which the graph opens.

#### How to Find the Leading Coefficient:
- Identify the term with the highest degree.
- The leading coefficient is the number multiplying this highest-degree term.

**Example**: In the polynomial \( f(x) = 4x^5 - 3x^3 + 2x^2 - 7 \), the term with the highest degree is \( 4x^5 \), so the leading coefficient is **4**.

### 3. **Example Problem**:

Given the polynomial \( f(x) = -2x^4 + 3x^2 - x + 5 \), find the degree and the leading coefficient.

#### Step 1: Find the Degree
The highest power of \( x \) is \( x^4 \), so the degree of the polynomial is **4**.

#### Step 2: Find the Leading Coefficient
The term with the highest degree is \( -2x^4 \), so the leading coefficient is **-2**.

Thus, for the polynomial \( f(x) = -2x^4 + 3x^2 - x + 5 \):
- Degree = **4**
- Leading Coefficient = **-2**

### 4. **Conclusion**:
- The **degree** of a polynomial is the highest exponent of \( x \) in the expression.
- The **leading coefficient** is the coefficient of the term with the highest degree.
Understanding these two features is crucial in determining the graph's behavior and overall structure of the polynomial.

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study