Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 4; zeros: 2, multiplicity 2; 3i
if r is a root (zero) , then x-r is a factor
factors ... (x - 2) (x - 2) (x + 3i) (x - 3i)
multiply the factors to form the polynomial
complex roots (with i in them) occur in conjugate pairs ... (x + i) & (x - i)
To form a polynomial with the given degree and zeros, we need to consider each zero and its multiplicity.
1. Zero 2 with multiplicity 2:
Since the zero 2 has a multiplicity of 2, it means that it appears twice in the factorization of the polynomial. Therefore, we can write this part of the polynomial as (x - 2)(x - 2).
2. Zero 3i:
Since 3i is a complex zero, its complex conjugate (-3i) would also be a zero. Therefore, we can write this part of the polynomial as (x - 3i)(x + 3i).
Now, we can combine the factors to form the polynomial:
f(x) = (x - 2)(x - 2)(x - 3i)(x + 3i).
To simplify the expression further, we can multiply the factors:
f(x) = (x - 2)(x - 2)(x^2 - (3i)^2)
= (x - 2)(x - 2)(x^2 + 9)
Expanding this expression:
f(x) = (x^2 - 4x + 4)(x^2 + 9)
= x^4 - 4x^3 + 4x^2 + 9x^2 - 36x + 36
= x^4 - 4x^3 + 13x^2 - 36x + 36
Therefore, the polynomial f(x) with real coefficients, degree 4, and zeros 2 (multiplicity 2) and 3i is:
f(x) = x^4 - 4x^3 + 13x^2 - 36x + 36.
To form the polynomial \( f(x) \) with the given zeros, we can use the fact that if \( a \) is a zero of a polynomial, then \( (x - a) \) is a factor of that polynomial.
Given the zeros 2 (with multiplicity 2) and 3i, we know that the factors of \( f(x) \) are \( (x - 2)^2 \) and \( (x - 3i) \).
Now, since \( f(x) \) has real coefficients, \( (x - 3i) \) must have a conjugate factor, which is \( (x + 3i) \).
Multiplying these factors together, we get:
\[
f(x) = (x - 2)^2 \cdot (x - 3i) \cdot (x + 3i)
\]
Expanding the expression, we have:
\[
f(x) = (x^2 - 4x + 4) \cdot (x^2 + 9)
\]
Finally, multiplying further:
\[
f(x) = x^4 - 4x^3 + 4x^2 + 9x^2 - 36x + 36
\]
Simplifying:
\[
f(x) = x^4 - 4x^3 + 13x^2 - 36x + 36
\]
So, the polynomial \( f(x) \) with the given zeros and real coefficients is \( x^4 - 4x^3 + 13x^2 - 36x + 36 \).