find the y-intercept of the polynomial function. f(x) = (x+1)(x-2)(x-1)^2

To find the **y-intercept** of the polynomial function \( f(x) = (x+1)(x-2)(x-1)^2 \), we need to substitute \( x = 0 \) into the function, as the y-intercept occurs when \( x = 0 \).

### Step-by-step calculation:
1. Substitute \( x = 0 \) into \( f(x) \):
   \[
   f(0) = (0+1)(0-2)(0-1)^2
   \]

2. Simplify each term:
   \[
   f(0) = (1)(-2)(-1)^2
   \]

3. Simplify the powers and multiply:
   \[
   f(0) = (1)(-2)(1) = -2
   \]

Thus, the **y-intercept** of the polynomial function is \( \mathbf{-2} \). 

Would you like further details or have any questions?

Here are some related questions:
1. How do you find the x-intercepts of a polynomial?
2. Can you explain how the degree of a polynomial affects its graph?
3. How do you factor higher-degree polynomials?
4. What are the turning points of this function?
5. How do you determine the end behavior of a polynomial?

**Tip**: The y-intercept of any function is always found by setting \( x = 0 \) in the equation.

Discover how to find the y-intercept of the polynomial function f(x) = (x+1)(x-2)(x-1)^2 by substituting x = 0 into the equation. This solution involves simplifying polynomials and calculating key intercepts. Suitable for Grades 9-11.

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