write a polynomial equation with integer coefficients that has the given roots y=3, y=-1, y=3

To construct a polynomial equation with integer coefficients that has the given roots \( y = 3, y = -1, y = 3 \), follow these steps:

### Step 1: Start with the root form
Given the roots:
- \( y = 3 \) (appears twice, so it's a repeated root)
- \( y = -1 \)

### Step 2: Construct factors from these roots
For each root \( r \), the corresponding factor is \( (y - r) \):
- For \( y = 3 \), the factor is \( (y - 3) \). Since it's a repeated root, the factor will be \( (y - 3)^2 \).
- For \( y = -1 \), the factor is \( (y + 1) \).

### Step 3: Combine the factors
The polynomial equation is the product of these factors:
\[
(y - 3)^2(y + 1)
\]

### Step 4: Expand the polynomial
First, expand \( (y - 3)^2 \):
\[
(y - 3)^2 = y^2 - 6y + 9
\]
Next, multiply this result by \( (y + 1) \):
\[
(y^2 - 6y + 9)(y + 1) = y^3 - 6y^2 + 9y + y^2 - 6y + 9
\]
Combine like terms:
\[
y^3 - 5y^2 + 3y + 9
\]

### Final Answer
The polynomial equation with integer coefficients is:
\[
y^3 - 5y^2 + 3y + 9 = 0
\]

Would you like more details or have any questions about this?

### Related Questions
1. How do you determine the degree of a polynomial from its roots?
2. What does it mean for a root to be "repeated" in a polynomial?
3. How do you factor a polynomial back from its expanded form?
4. Can a polynomial with real coefficients have non-integer roots?
5. What is the relationship between the coefficients of a polynomial and its roots?

### Tip
The multiplicity of a root refers to how many times that root appears in the factorization of the polynomial.

Learn how to construct a polynomial equation with integer coefficients that has the roots y = 3 and y = -1, with a repeated root at y = 3. This example expands on core algebraic concepts like polynomials and root factorization, suitable for students in Grades 9-12.

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