Write a polynomial of the lowest degree with real coefficients and with zeros 6-3i (multiplicity 1) and 0 ( multiplicity 5)
complex roots (zeros with i) occur in conjugate pairs
f(x) = (x-6+3i)(x-6-3i)(x^5)
To find the polynomial, we need to consider all the zeros and their multiplicities.
First, we have the zero 6-3i with multiplicity 1. This means that our polynomial will have a factor of (x - (6-3i)). Since we want real coefficients, we also need to include the complex conjugate of this factor, which is (x - (6+3i)).
Next, we have the zero 0 with multiplicity 5. This means our polynomial will have a factor of (x - 0)^5, which simplifies to x^5.
Now we can construct our polynomial by multiplying all the factors together:
(x - (6-3i))(x - (6+3i))(x^5)
Expanding the first two factors, we have:
((x-6) + 3i)((x-6) - 3i)(x^5)
Using the difference of squares, we simplify further:
((x-6)^2 - (3i)^2)(x^5)
Simplifying the squares of the complex numbers:
((x^2 - 12x + 36) + 9)(x^5)
Expanding further:
(x^2 - 12x + 45)(x^5)
Finally, multiplying the remaining factors together, we have our polynomial:
x^7 - 12x^6 + 45x^5 - 12x^3 + 144x^2 - 540x + 2025
So, the polynomial of the lowest degree with real coefficients and with zeros 6-3i (multiplicity 1) and 0 (multiplicity 5) is:
f(x) = x^7 - 12x^6 + 45x^5 - 12x^3 + 144x^2 - 540x + 2025
To write a polynomial with these given zeros, we need to consider the fact that complex roots occur in conjugate pairs. The complex root 6-3i will have its conjugate as well: 6+3i. Therefore, the polynomial will have the factors (x - 6 + 3i) and (x - 6 - 3i).
To find the polynomial, we multiply these factors together:
(x - 6 + 3i)(x - 6 - 3i)
Using the difference of squares, this product simplifies to:
(x - 6)^2 - (3i)^2
Expanding further:
(x - 6)^2 - 9i^2
Since the given question asks for a polynomial with real coefficients, we need to eliminate the imaginary part. Recognize that i^2 is equal to -1:
(x - 6)^2 - 9(-1)
Simplifying further:
(x - 6)^2 + 9
Now we have the polynomial in the standard form. To account for the zero 0 with a multiplicity of 5, we multiply this polynomial with (x - 0)^5:
(x - 0)^5 * [(x - 6)^2 + 9]
Expanding:
x^5 * [(x - 6)^2 + 9]
To simplify, you could also expand the square:
x^5 * [x^2 - 12x + 36 + 9]
Combining like terms:
x^7 - 12x^6 + 45x^5 - 108x^4 + 81x^3
So the polynomial of the lowest degree with real coefficients and the given zeros is:
x^7 - 12x^6 + 45x^5 - 108x^4 + 81x^3