Use the equation of the polynomial function ​f(x)equalsleft parenthesis x minus 1 right parenthesis left parenthesis x squared minus 4 right parenthesis to complete the following.

Question

​(a) State the degree and the leading coefficient.
​(b) Describe the end behavior of the graph of the function.
​(c) Support your answer by graphing the function.
Question content area bottom
Part 1
​(a) The degree of the polynomial is
  
enter your response here and the leading coefficient is
  
enter your response here.

Answer 1

To analyze the polynomial function \( f(x) = (x - 1)(x^2 - 4) \), we'll first start with the necessary calculations.

Part 1: Degree and Leading Coefficient

(a) Finding the degree and leading coefficient:

1. The polynomial can be rewritten as: \[ f(x) = (x - 1)(x^2 - 4) \] The term \( (x^2 - 4) \) can be factored further, but here we can still analyze it in its product form.

2. The degree of each polynomial factor is:

   • \( (x - 1) \) has a degree of 1.
   • \( (x^2 - 4) \) has a degree of 2.

3. To find the overall degree of the polynomial, we add the degrees of the factors: \[ \text{Degree of } f(x) = 1 + 2 = 3. \]

4. The leading coefficient comes from multiplying the leading coefficients of each factor. The leading coefficient of \( (x - 1) \) is 1, and the leading coefficient of \( (x^2 - 4) \) is also 1. Therefore, the leading coefficient of \( f(x) \) is: \[ 1 \cdot 1 = 1. \]

Thus, the degree of the polynomial is 3, and the leading coefficient is 1.

Part 2: End Behavior

(b) Describing the end behavior of the graph of the function:

• Since the degree of the polynomial is odd (3) and the leading coefficient is positive (1), we can describe the end behavior as follows:
  • As \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
  • As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).

Part 3: Graphing the Function

(c) To graph the function, we can find critical points, such as the roots:

1. Roots of the function: Set \( f(x) = 0 \): \[ f(x) = (x - 1)(x^2 - 4) = 0. \]

   This gives us:

   • \( x - 1 = 0 \) ⇒ \( x = 1 \)
   • \( x^2 - 4 = 0 \) ⇒ \( x^2 = 4 \) ⇒ \( x = 2 \) or \( x = -2 \)

   Therefore, the roots are \( x = 1, 2, -2 \).

2. Sketching the graph:

   • Mark the points where the function crosses the x-axis: \( (-2, 0), (1, 0), (2, 0) \).
   • Since the leading coefficient is positive, the graph will rise to the right and fall to the left.
   • Analyze typical points or the y-intercept to get a better graph shape (e.g., evaluate \( f(0) \)): \[ f(0) = (0 - 1)(0^2 - 4) = (-1)(-4) = 4. \] So, the y-intercept is (0, 4).

Using this information, you can sketch the function, ensuring it crosses the x-axis at the identified points and reflects the described end behavior.

In conclusion:

• (a) Degree: 3, Leading coefficient: 1.
• (b) End behavior: Rises to \( +\infty \) as \( x \to +\infty \); falls to \( -\infty \) as \( x \to -\infty \).