Question

Form a polynomial f(x) with real coefficients having the given degree and zeros.

Degree 4; zeros: 6 (Multiplicity 2); 3i
Enter the expanded polynomial. Let a represent the leading coefficient.
f(x) = a( )

Answer 1

The polynomial f(x) with real coefficients having the given degree and zeros is:

f(x) = a(x - 6)^2(x - 3i)(x + 3i)

To expand this polynomial, we can use the fact that (a + b)(a - b) = a^2 - b^2. Substituting a = x - 6 and b = 3i, we get:

(x - 6 + 3i)(x - 6 - 3i) = (x - 6)^2 + 9

Therefore, the expanded polynomial is:

f(x) = a(x - 6)^2(x - 3i)(x + 3i)
f(x) = a(x - 6)^2(x^2 + 9)
f(x) = a(x^4 - 12x^3 + 57x^2 - 108x + 81)

So, the expanded polynomial is f(x) = a(x^4 - 12x^3 + 57x^2 - 108x + 81).

Answer 2

Explanation:

The polynomial f(x) with real coefficients, degree 4, and zeros 6 (multiplicity 2), 3i can be formed as follows:

Since the zero 6 has a multiplicity of 2, it appears twice in the factored form of f(x), i.e., (x-6)(x-6) = (x-6)^2.

The other zero is 3i, which means its complex conjugate, -3i, is also a zero. Therefore, the factored form of f(x) can be written as:

(x-6)^2(x-3i)(x+3i)

Expanding this expression, we get:

f(x) = (x-6)^2(x^2 + 9)

Multiplying this out, we get:

f(x) = x^4 - 12x^3 + 81x^2 - 216x + 324

Therefore, the polynomial f(x) with real coefficients, degree 4, and zeros 6 (multiplicity 2), 3i can be written as:

f(x) = x^4 - 12x^3 + 81x^2 - 216x + 324.