Question

I'm studying polynomial functions and I need help with a problem. Find the equation of a polynomial function with roots of -2, 1 (order 2), and 3, and has a y-intercept of 12. Can you find the solution to this problem on the website?.

Answer 1

To find the equation of a polynomial function given its roots and a y-intercept, we can follow these steps:

1. **Identify the roots**: The given roots are -2, 1 (with order 2), and 3. This means that the polynomial can be expressed in the form:
\[
f(x) = k(x + 2)(x - 1)^2(x - 3)
\]
where \( k \) is a constant that we need to find.

2. **Expand the polynomial**:
- Start with the factors:
- The factor for the root -2 is \( (x + 2) \).
- The factor for the root 1 with order 2 is \( (x - 1)^2 = (x - 1)(x - 1) \).
- The factor for the root 3 is \( (x - 3) \).
- The polynomial can now be expressed as:
\[
f(x) = k(x + 2)(x - 1)^2(x - 3)
\]

3. **Find the y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
f(0) = k(0 + 2)(0 - 1)^2(0 - 3) = k(2)(1)(-3) = -6k
\]
We know that the y-intercept is 12, so we set up the equation:
\[
-6k = 12
\]

4. **Solve for \( k \)**:
\[
k = \frac{12}{-6} = -2
\]

5. **Write the final polynomial**:
Substitute \( k \) back into the polynomial:
\[
f(x) = -2(x + 2)(x - 1)^2(x - 3)
\]

6. **Expand the polynomial (optional)**:
If you want the polynomial in standard form, you can expand it further. First, expand \( (x - 1)^2 \):
\[
(x - 1)^2 = x^2 - 2x + 1
\]
Now substitute this back in:
\[
f(x) = -2(x + 2)(x^2 - 2x + 1)(x - 3)
\]
You can expand it step-by-step to get it in standard form, but the key factorization is enough for most purposes.

In summary, the polynomial function you are looking for is:
\[
f(x) = -2(x + 2)(x - 1)^2(x - 3)
\]