Find a polynomial of lowest degree with only real coefficients and having the given zeros.
-2+i, -2-i, 3, -3
Zeros are also "x = ..." statements, so x = -2 + i, x = -2 - i, etc...
Since you need REAL coefficients in the polynomial, change the x = -2 + i and x = -2 - i statements to "clean" polynomials, like this:
x = -2 + i
x + 2 = i (square each side to get rid of the i)
(x+2)^2 = i^2
x^2 + 4x + 4 = -1
==> x^2 + 4x + 5 = 0
x = -2 - i
(x+2)^2 = (-i)^2
I'm not going to go further with this because you get the same exact equation as the one before. So now when you multiply all the zeroes together, you get this:
0 = (x^2 + 4x + 5)(x-3)(x+3)
Multiply it out and you get:
x^4 + 4x^3 - 4x^2 - 36x - 45 = 0
Does this make sense?
The polynomial would have to have 4 zeros, meaning it would have to be a polynomial of the 4th degree. The general form for a polynomial of the 4th degree with zeros a, b, c and d would be:
f*(x-a)*(x-b)*(x-c)*(x-d)
where f is a random real number (lets take this to be 1 in this case).
So, if we fill in the zeros you were given we get that:
(x-(-2+i))*(x-(-2-i))*(x-3)*(x+3) =
(x+2-i))*(x+2+i))*(x-3)*(x+3) =
When we multiply the first two factors and the last two, we get:
(x^2 + 2x + ix + 2x + 4 + 2i -ix - 2i +1) * (x^2 - 9) =
(x^2 + 4x + 5)*(x^2 - 9) =
(x^4 - 9x^2 + 4x^3 - 36x + 5x^2 -45) =
x^4 + 4x^3 - 4x^2 -36x - 45
this is a polynomial of the 4th degree which has the given values as its zeros
To find a polynomial with the given zeros, we can use the property of complex roots. If a polynomial has a complex root, then its conjugate is also a root.
Given the zeros: -2+i, -2-i, 3, -3
Since -2+i is a root, its conjugate -2-i is also a root.
So, the roots are: -2+i, -2-i, 3, -3
To find the polynomial, we can set up the factors using these roots:
(x - (-2+i))(x - (-2-i))(x - 3)(x + 3) = 0
Simplifying this equation, we get:
(x + 2 - i)(x + 2 + i)(x - 3)(x + 3) = 0
Now, let's multiply these factors to get the polynomial:
(x^2 + 4x + 4 - ix - 2i + ix + 2i - i^2)(x^2 - 9) = 0
Simplifying further:
(x^2 + 4x + 4 - 1)(x^2 - 9) = 0
(x^2 + 4x + 3)(x - 3)(x + 3) = 0
Finally, we can expand this expression to get the polynomial:
(x^2 + 4x + 3)(x^2 - 9) = 0
(x^4 - 9x^2 + 4x^3 - 36x + 3x^2 - 27) = 0
(x^4 + 4x^3 - 6x^2 - 36x - 27) = 0
Therefore, a polynomial of lowest degree with only real coefficients and having the given zeros is:
f(x) = x^4 + 4x^3 - 6x^2 - 36x - 27
To find the polynomial of lowest degree with only real coefficients and having the given zeros, we can use the fact that complex roots occur in conjugate pairs.
Given the zeros: -2+i, -2-i, 3, -3
First, let's consider the zeros -2+i and -2-i. Since they are complex conjugates, it means that their sum is a real number and their product is a real number as well.
The sum of -2+i and -2-i is -4, which is a real number. The product of -2+i and -2-i can be found by using the difference of squares formula: (a + b)(a - b) = a^2 - b^2. In this case, a = -2 and b = i. Therefore, the product is (-2)^2 - i^2 = 4 - (-1) = 5.
Now, let's write the equation for the polynomial using these complex roots:
(x - (-2+i))(x - (-2-i))(x - 3)(x - (-3)) = 0
Simplifying this equation, we get:
(x + 2 - i)(x + 2 + i)(x - 3)(x + 3) = 0
Expanding these factors, we get:
(x^2 + (2 - i)x + (2 + i)x + (2 + i)(2 - i))(x^2 - 9) = 0
Simplifying further, we have:
(x^2 + 2x - ix + 2x + 4 + 2i - ix - i + ix - i^2)(x^2 - 9) = 0
Combining like terms and simplifying:
(x^2 + 4x + 4 + i^2)(x^2 - 9) = 0
(x^2 + 4x + 4 - 1)(x^2 - 9) = 0
(x^2 + 4x + 3)(x^2 - 9) = 0
Now, simplify each binomial using FOIL:
(x + 1)(x + 3)(x + 3)(x - 3) = 0
Finally, multiply the binomials together:
(x + 1)(x^2 + 6x + 9)(x - 3) = 0
Expanding the remaining factors:
(x^3 + 6x^2 + 9x + x^2 + 6x + 9x + 9)(x - 3) = 0
Combining like terms:
(x^3 + 7x^2 + 18x + 9)(x - 3) = 0
Expanding one final time:
x^4 - 3x^3 + 7x^3 - 21x^2 + 18x^2 - 54x + 9x - 27 = 0
Combining like terms:
x^4 + 4x^3 - 3x^2 - 45x - 27 = 0
Therefore, the polynomial of lowest degree with only real coefficients and having the given zeros is:
f(x) = x^4 + 4x^3 - 3x^2 - 45x - 27