Question

What are the degree, type, leading coefficient, and constant term of each polynomial function? a) \(f(x)=-x+3\) b) \(y=9 x^{2}\) \(g(x)=3 x^{4}+3 x^{2}-2 x+1\) d) \(k(x)=4-3 x^{3}\) e) \(y=-2 x^{5}-2 x^{3}+9\) f) \(h(x)=-6\)

Short answer

a) Degree: 1, Type: Linear, Leading Coefficient: -1, Constant: 3. b) Degree: 2, Type: Quadratic, Leading Coefficient: 9, Constant: 0. c) Degree: 4, Type: Polynomial (4), Leading Coefficient: 3, Constant: 1. d) Degree: 3, Type: Cubic, Leading Coefficient: -3, Constant: 4. e) Degree: 5, Type: Polynomial (5), Leading Coefficient: -2, Constant: 9. f) Degree: 0, Type: Constant, Leading Coefficient: -6, Constant: -6.

Step-by-step solution

Identify the Degree of Each Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. For each given polynomial function, find the term with the highest exponent and note its degree. - For (a) \( f(x) = -x + 3 \), the highest power of x is 1. Degree: 1. - For (b) \( y = 9x^2 \), the highest power of x is 2. Degree: 2. - For (c) \( g(x) = 3x^4 + 3x^2 - 2x + 1 \), the highest power of x is 4. Degree: 4. - For (d) \( k(x) = 4 - 3x^3 \), the highest power of x is 3. Degree: 3. - For (e) \( y = -2x^5 - 2x^3 + 9 \), the highest power of x is 5. Degree: 5. - For (f) \( h(x) = -6 \), the highest power of x is 0 (no variable term). Degree: 0.

Identify the Type of Each Polynomial

The type of a polynomial can be characterized by the degree: - Degree 0: Constant - Degree 1: Linear - Degree 2: Quadratic - Degree 3: Cubic - Degree 4 and above: Polynomial of degree n - For (a) \( f(x) = -x + 3 \), type: Linear. - For (b) \( y = 9 x^2 \), type: Quadratic. - For (c) \( g(x) = 3x^4 + 3x^2 - 2x + 1 \), type: Polynomial of degree 4. - For (d) \( k(x) = 4 - 3x^3 \), type: Cubic. - For (e) \( y = -2x^5 - 2x^3 + 9 \), type: Polynomial of degree 5. - For (f) \( h(x) = -6 \), type: Constant.

Identify the Leading Coefficient of Each Polynomial

The leading coefficient is the coefficient of the term with the highest degree. - For (a) \( f(x) = -x + 3 \), leading coefficient: -1. - For (b) \( y = 9x^2 \), leading coefficient: 9. - For (c) \( g(x) = 3x^4 + 3x^2 - 2x + 1 \), leading coefficient: 3. - For (d) \( k(x) = 4 - 3x^3 \), leading coefficient: -3. - For (e) \( y = -2x^5 - 2x^3 + 9 \), leading coefficient: -2. - For (f) \( h(x) = -6 \), leading coefficient: -6 (since degree is 0, the constant term is also the leading coefficient).

Identify the Constant Term of Each Polynomial

The constant term is the term without the variable x. - For (a) \( f(x) = -x + 3 \), constant term: 3. - For (b) \( y = 9x^2 \), constant term: 0 (no constant term). - For (c) \( g(x) = 3x^4 + 3x^2 - 2x + 1 \), constant term: 1. - For (d) \( k(x) = 4 - 3x^3 \), constant term: 4. - For (e) \( y = -2x^5 - 2x^3 + 9 \), constant term: 9. - For (f) \( h(x) = -6 \), constant term: -6.

Key concepts

Degree of Polynomial

The degree of a polynomial is a crucial concept to understand. It refers to the highest power of the variable x in the polynomial expression. This power (or exponent) essentially tells us the polynomial's 'complexity' and helps categorize it.
For example:
- For the polynomial \(f(x) = -x + 3\), the highest power of x is 1. So, its degree is 1.
- Similarly, in the polynomial \(y = 9x^2\), the highest power of x is 2, making the degree 2.
The degree determines the shape of the graph of the polynomial and significantly affects its behavior.
Here's a quick cheat sheet for polynomial degrees:

• Degree 0: Constant
• Degree 1: Linear
• Degree 2: Quadratic
• Degree 3: Cubic
• Degree 4 and higher: Polynomial of degree n

As another example, the polynomial \(g(x) = 3x^4 + 3x^2 - 2x + 1\) has a degree of 4 because the highest power of x is 4.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. This number is very important because it influences the direction and the steepness of the polynomial graph.
For example:

• In the polynomial \(f(x) = -x + 3\), the leading coefficient is -1 because it is the coefficient of the \(x\) term, which has the highest degree (1).
• The polynomial \(y = 9x^2\) has a leading coefficient of 9, as the coefficient of the \(x^2\) term (highest degree 2) is 9.

Identifying the leading coefficient is straightforward: just look for the term with the highest exponent, and its coefficient is the leading one. This helps in analyzing how the polynomial behaves as the values of x become very large or very small.
For instance, in the polynomial \(g(x) = 3x^4 + 3x^2 - 2x + 1\), the leading coefficient is 3, corresponding to the \(x^4\) term.

Constant Term

The constant term of a polynomial is the number that stands alone, without any variable attached to it. This term represents the value of the polynomial when all the variables are equal to zero.
For example:

• In the polynomial \(f(x) = -x + 3\), the constant term is 3 because it remains when \(x\) is zero.
• For \(y = 9x^2\), there is no constant term, so it is considered 0.

The constant term is particularly useful in determining the polynomial's y-intercept, which is the point where the graph of the polynomial crosses the y-axis. Simply put, when the variable x is set to zero, the constant term is the value you get.
Another example is the polynomial \(g(x) = 3x^4 + 3x^2 - 2x + 1\), where the constant term is 1.

Polynomial Type

Polynomials can be categorized into various types based on their degree, which is the highest power of their variable.
The main types include:

• Constant Polynomial: Degree 0, as in \(h(x) = -6\).
• Linear Polynomial: Degree 1, like \(f(x) = -x + 3\).
• Quadratic Polynomial: Degree 2, such as \(y = 9x^2\).
• Cubic Polynomial: Degree 3, like \(k(x) = 4 - 3x^3\).
• Quartic Polynomial: Degree 4, such as \(g(x) = 3x^4 + 3x^2 - 2x + 1\).
• Quintic Polynomial: Degree 5, like \(y = -2x^5 - 2x^3 + 9\).

Understanding the type of a polynomial helps in identifying its general shape, properties, and how it behaves for large or small values of x. This categorization builds the foundation for more advanced topics in algebra and calculus.