find a polynomial f(x) of degree 3 with real coefficients and the zeros of -1, -1+i

To find a polynomial \( f(x) \) of degree 3 with real coefficients and given zeros \(-1\) and \(-1+i\), we need to remember that the complex zeros of polynomials with real coefficients come in conjugate pairs. Therefore, the zeros of the polynomial are \(-1\), \(-1+i\), and \(-1-i\).

Given these zeros, we can express the polynomial as the product of the factors corresponding to these zeros.

The zeros \(-1\), \(-1+i\), and \(-1-i\) give us the following factors:
- \( (x + 1) \) from the zero \(-1\)
- \( (x + 1 - i) \) from the zero \(-1+i\)
- \( (x + 1 + i) \) from the zero \(-1-i\)

The polynomial \( f(x) \) is obtained by multiplying these factors:

\[ f(x) = (x + 1)(x + 1 - i)(x + 1 + i) \]

First, let's simplify the product of the two complex conjugate factors:

\[ (x + 1 - i)(x + 1 + i) \]

Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \), we get:

\[ (x + 1)^2 - i^2 \]

Since \( i^2 = -1 \), this becomes:

\[ (x + 1)^2 - (-1) \]
\[ (x + 1)^2 + 1 \]

Now, expand \( (x + 1)^2 \):

\[ (x + 1)^2 = x^2 + 2x + 1 \]

Therefore,

\[ (x + 1)^2 + 1 = x^2 + 2x + 1 + 1 = x^2 + 2x + 2 \]

So, the polynomial can now be written as:

\[ f(x) = (x + 1)(x^2 + 2x + 2) \]

Next, we expand this product:

\[ f(x) = x(x^2 + 2x + 2) + 1(x^2 + 2x + 2) \]

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

Combine like terms:

\[ f(x) = x^3 + 3x^2 + 4x + 2 \]

Thus, the polynomial \( f(x) \) of degree 3 with the given zeros is:

\[ f(x) = x^3 + 3x^2 + 4x + 2 \]

Do you want details or have any questions?

Here are 8 questions related to this topic:
1. How do you determine the conjugate of a complex number?
2. What is the significance of the difference of squares formula?
3. How do you multiply polynomials?
4. Can you explain why complex zeros of polynomials with real coefficients come in conjugate pairs?
5. How do you find the zeros of a polynomial?
6. What is the general form of a cubic polynomial?
7. How can you verify if a given polynomial has specific zeros?
8. What are some applications of polynomials in real life?

**Tip:** When multiplying polynomials, carefully align and combine like terms to ensure accuracy in your final expression.

Learn how to find a polynomial f(x) of degree 3 with real coefficients and given zeros -1, -1+i, and -1-i. Detailed step-by-step solution with explanations of polynomial multiplication, complex conjugates, and the difference of squares formula. Suitable for high school students studying algebra and polynomial functions.

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