Choose a correct statement about the function,

f(x)=(x−1)(x2−x−6).

 A.

f(x) has degree 2 and y-intercept of 6
 
 B.

f(x) has degree 3 and y-intercept of −6
 
 C. 

None

 D.

f(x) has undefined degree and y-intercept of −6
 
 E.

f(x) has degree 3 and y-intercept of 6
 

To choose the correct statement about the function:

\[
f(x) = (x - 1)(x^2 - x - 6)
\]

we need to determine:

1. The degree of the function \( f(x) \).
2. The y-intercept of the function, which is the value of \( f(x) \) when \( x = 0 \).

### Step 1: Determine the Degree of the Function

To find the degree of \( f(x) \), we need to expand the expression:

\[
f(x) = (x - 1)(x^2 - x - 6)
\]

Multiply the terms:

\[
f(x) = x(x^2 - x - 6) - 1(x^2 - x - 6)
\]

\[
f(x) = (x^3 - x^2 - 6x) - (x^2 - x - 6)
\]

Combine like terms:

\[
f(x) = x^3 - 2x^2 - 5x + 6
\]

The highest power of \( x \) in the expanded form is \( x^3 \), so the degree of the function is **3**.

### Step 2: Find the y-intercept of the Function

The y-intercept is the value of \( f(x) \) when \( x = 0 \):

\[
f(0) = (0 - 1)(0^2 - 0 - 6)
\]

\[
f(0) = (-1)(-6) = 6
\]

So, the y-intercept of the function is **6**.

### Conclusion

The correct statement is:

**E. \( f(x) \) has degree 3 and y-intercept of 6.**

Would you like further details or have any questions?

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### Related Questions

1. What is the degree of a polynomial function, and how is it determined?
2. How do you find the y-intercept of a function?
3. Why is the highest power of \( x \) important in determining the function's behavior?
4. How does expanding a polynomial help in finding its degree?
5. What does the y-intercept tell you about a function graphically?

**Tip:** To find the y-intercept of a polynomial, simply substitute \( x = 0 \) and evaluate the function.

Learn about the degree and y-intercept of the polynomial function f(x) = (x - 1)(x^2 - x - 6). Understand how to determine the degree of a polynomial and find its y-intercept with a detailed step-by-step solution.

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