To form a polynomial $f(x)$ with real coefficients given the specified degree and zeros, let's follow these steps based on the information provided:

Given Information:

• Degree: 4
• Zeros: $3 + 3i$ and $3$ with multiplicity 2

Step 1: Understand Complex Conjugate Pairs

Since the polynomial has real coefficients, if $3 + 3i$ is a zero, then its complex conjugate $3 - 3i$ must also be a zero.

Step 2: List all Zeros

The zeros of the polynomial are:

• $3 + 3i$
• $3 - 3i$
• $3$ (with multiplicity 2)

Step 3: Form the Polynomial

The polynomial can be expressed as the product of factors corresponding to its zeros: $f(x) = a(x - (3 + 3i))(x - (3 - 3i))(x - 3)^2$

Step 4: Multiply the Factors

1. Multiply the complex conjugate factors: $(x - (3 + 3i))(x - (3 - 3i)) = (x - 3 - 3i)(x - 3 + 3i)$ This simplifies using the difference of squares formula: $(x - 3)^2 - (3i)^2 = (x - 3)^2 + 9$ Expanding $(x - 3)^2$: $(x - 3)^2 = x^2 - 6x + 9$ So, we have: $(x - 3)^2 + 9 = x^2 - 6x + 9 + 9 = x^2 - 6x + 18$

2. Now, combine this result with the other factor $(x - 3)^2$: $f(x) = a(x^2 - 6x + 18)(x - 3)^2$

3. Expand $(x - 3)^2$ again: $(x - 3)^2 = x^2 - 6x + 9$

4. Multiply $(x^2 - 6x + 18)(x^2 - 6x + 9)$ to get the expanded form: $f(x) = a(x^4 - 12x^3 + 63x^2 - 108x + 162)$

Final Answer

The polynomial $f(x)$ in its expanded form is: $f(x) = a(x^4 - 12x^3 + 63x^2 - 108x + 162)$

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Relative Questions:

1. How do you determine the conjugate of a complex number?
2. What is the significance of the multiplicity of a root in a polynomial?
3. How do you expand expressions using the distributive property?
4. What is the difference between real and complex roots in a polynomial?
5. How can the degree of a polynomial affect its possible number of roots?

Tip:

Always remember that if a polynomial has real coefficients, complex roots must come in conjugate pairs.