Write a polynomial of least degree with integer coefficients that has the given zeros.

-3,-1+ square root of 3

Leave your answer in factored form, but make sure there are no irrational or imaginary values.

To construct a polynomial with the given zeros, we need to account for both the given roots and any necessary conjugate roots. The given zeros are \(-3\) and \(-1 + \sqrt{3}\).

### Step 1: Include the Conjugate Root
For the zero \(-1 + \sqrt{3}\), its conjugate \(-1 - \sqrt{3}\) must also be a root of the polynomial to ensure that the coefficients of the polynomial are integers.

### Step 2: Write the Factors
The polynomial will have factors corresponding to each root:
- For the root \(-3\), the factor is \((x + 3)\).
- For the roots \(-1 + \sqrt{3}\) and \(-1 - \sqrt{3}\), the factors are \((x + 1 - \sqrt{3})\) and \((x + 1 + \sqrt{3})\).

### Step 3: Multiply the Conjugate Factors
First, let's multiply the conjugate factors:
\[
(x + 1 - \sqrt{3})(x + 1 + \sqrt{3})
\]

This is a difference of squares:
\[
(x + 1)^2 - (\sqrt{3})^2
\]
\[
= (x + 1)^2 - 3
\]
\[
= x^2 + 2x + 1 - 3
\]
\[
= x^2 + 2x - 2
\]

### Step 4: Multiply All Factors
Now, multiply this result by the factor corresponding to the root \(-3\):
\[
(x + 3)(x^2 + 2x - 2)
\]

Leave the polynomial in factored form:
\[
\boxed{(x + 3)(x^2 + 2x - 2)}
\]

This is the polynomial of least degree with integer coefficients that has the given zeros.

Would you like to see the polynomial expanded or have any further questions?

Here are 5 related questions:
1. How do you determine the conjugate of a root that involves square roots?
2. What is the difference between a polynomial in factored form and expanded form?
3. Can you explain why the product of conjugates eliminates irrational numbers?
4. What does the "least degree" of a polynomial mean?
5. How would the process change if one of the roots were imaginary?

**Tip:** Always remember to include conjugate pairs when forming polynomials with real coefficients to eliminate any irrational parts.

Write a polynomial of least degree with integer coefficients that has the given zeros: -3, -1 + sqrt(3). Leave your answer in factored form, but make sure there are no irrational or imaginary values.

Learn how to write a polynomial of least degree with integer coefficients given the zeros -3 and -1 + sqrt(3). The solution includes the conjugate root theorem and the difference of squares formula, providing a clear step-by-step process. Suitable for students in Grades 10-12.

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