Find integers A and C and D such that the equation Ax4+0x3+Cx^2+DX+0 has 2-i√5 as a root.

To find integers \( A \), \( C \), and \( D \) such that the polynomial \( A x^4 + 0 x^3 + C x^2 + D x + 0 = 0 \) has \( 2 - i \sqrt{5} \) as a root, we proceed step by step:

### Step 1: Use the fact that if \( 2 - i \sqrt{5} \) is a root, its conjugate \( 2 + i \sqrt{5} \) must also be a root.
This is because the coefficients of the polynomial are real, and complex roots always come in conjugate pairs.

### Step 2: Form a quadratic factor.
Given that \( 2 - i \sqrt{5} \) and \( 2 + i \sqrt{5} \) are roots, a quadratic polynomial with these roots is:

\[
(x - (2 - i \sqrt{5}))(x - (2 + i \sqrt{5}))
\]

### Step 3: Expand the product.
First, simplify the expression:

\[
(x - 2 + i \sqrt{5})(x - 2 - i \sqrt{5}) = \left( (x - 2) - i \sqrt{5} \right) \left( (x - 2) + i \sqrt{5} \right)
\]

This is a difference of squares:

\[
= (x - 2)^2 - (i \sqrt{5})^2
\]

Since \( (i \sqrt{5})^2 = -5 \), we get:

\[
= (x - 2)^2 + 5
\]

### Step 4: Expand \( (x - 2)^2 + 5 \).
First, expand \( (x - 2)^2 \):

\[
(x - 2)^2 = x^2 - 4x + 4
\]

Now add 5:

\[
x^2 - 4x + 4 + 5 = x^2 - 4x + 9
\]

So the quadratic factor corresponding to the roots \( 2 - i \sqrt{5} \) and \( 2 + i \sqrt{5} \) is:

\[
x^2 - 4x + 9
\]

### Step 5: Use the fact that the polynomial is quartic.
Since the polynomial is of degree 4, there must be two more roots. A natural choice is to assume that these additional roots are \( 0 \) and \( 0 \) (since the constant term in the polynomial is 0). This gives the factor \( x^2 \).

### Step 6: Write the complete polynomial.
The complete polynomial is the product of \( x^2 \) and \( x^2 - 4x + 9 \):

\[
x^2(x^2 - 4x + 9) = x^4 - 4x^3 + 9x^2
\]

Thus, comparing this with the given form \( A x^4 + 0 x^3 + C x^2 + D x + 0 \), we have:

\[
A = 1, \quad C = 9, \quad D = 0
\]

### Final Answer:
The integers are:
\[
A = 1, \quad C = 9, \quad D = 0
\]

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Would you like more details or have any questions about the steps? Here are some related questions:

1. What happens when complex conjugate roots are involved in a polynomial?
2. How do we handle roots when coefficients are real but some roots are complex?
3. Can we generalize this process to higher-degree polynomials?
4. Why do complex roots appear in conjugate pairs for polynomials with real coefficients?
5. What are the implications of having zero as a repeated root in a polynomial?

**Tip:** When expanding expressions with complex conjugates, use the difference of squares formula to simplify your calculations.

Find integers A and C and D such that the equation Ax^4 + 0x^3 + Cx^2 + Dx + 0 has 2 - i√5 as a root.

Discover how to find integers A, C, and D for the quartic equation Ax^4 + 0x^3 + Cx^2 + Dx + 0 given that 2 - i√5 is a root. This solution explores key algebraic concepts like complex conjugates and the conjugate root theorem, suitable for students in Grades 10-12.

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