since complex roots occur in conjugate pairs,
(x-3)^2 (x-(5+i)) (x-(5-i))
3(multiplicity 2), 5+i(multiplicity 1)
(x-3)^2 (x-(5+i)) (x-(5-i))
Let's start by writing the factors corresponding to each zero and multiplicity:
For 3 (multiplicity 2), we have (x-3)(x-3).
For 5+i (multiplicity 1) and its conjugate 5-i (multiplicity 1), we have (x - (5+i))(x - (5-i)).
Next, let's multiply these factors out to find the polynomial in factored form:
(x - 3)(x - 3)(x - (5+i))(x - (5-i))
Expanding this expression yields:
(x - 3)(x - 3)(x - 5 - i)(x - 5 + i)
Next, let's simplify this expression:
(x^2 - 6x + 9)(x^2 - 10x + 26)
Now, let's multiply the binomials:
(x^2)(x^2) + (x^2)(-10x) + (x^2)(26) + (-6x)(x^2) + (-6x)(-10x) + (-6x)(26) + (9)(x^2) + (9)(-10x) + (9)(26)
Simplifying, we get:
x^4 - 16x^3 + 80x^2 - 156x + 234
Therefore, the polynomial function of minimum degree with real coefficients whose zeros are 3 (multiplicity 2) and 5+i (multiplicity 1) is:
f(x) = x^4 - 16x^3 + 80x^2 - 156x + 234, in standard form.
First, we know that the zero 3 has a multiplicity of 2. This means that (x - 3) appears twice as a factor in our polynomial.
Next, we know that the zero 5+i has a multiplicity of 1. This means that (x - (5+i)) appears once as a factor in our polynomial. To match the requirement of having real coefficients, we can assume that the complex conjugate, 5-i, also has multiplicity 1. Therefore, (x - (5-i)) should be another factor.
Now, we can combine all these factors to obtain the polynomial function:
(x - 3)(x - (5+i))(x - (5-i))
To simplify this expression, we can expand the polynomial and express it in standard form:
(x - 3)(x - 5 - i)(x - 5 + i)
Using the distributive property, we can expand the expression:
(x - 3)(x^2 - 5x + xi - 5x + 25 + 5i - xi + 5i - i^2)
Simplifying further:
(x - 3)(x^2 - 10x + 25 + 1)
(x - 3)(x^2 - 10x + 26)
Now, let's multiply the factors:
x(x^2 - 10x + 26) - 3(x^2 - 10x + 26)
x^3 - 10x^2 + 26x - 3x^2 + 30x - 78
Finally, we collect like terms and express the polynomial in standard form:
x^3 - 13x^2 + 56x - 78
Therefore, the polynomial function of minimum degree with real coefficients and the given zeros and multiplicities is:
f(x) = x^3 - 13x^2 + 56x - 78