write a fourth degree polynomial function a leading coefficient o 1 given three of its zeros are -1, 3 and 2i, then re-write it in simplified form.
The key concept here is that imaginary numbers always appear as conjugates
so if 2i is a zero, so is -2a
so we would have x = ±2i
x^2 = 4i^2
x^2 = -4
so x^2 + 4 is a factor
the polynomial is
f(x) = (x+1)(x-3)(x^2 + 4)
I will leave it up to you to expand it, but I would consider the factored form more simplified than the expanded form.
To find the fourth-degree polynomial with given zeros -1, 3, and 2i, you also need to consider the complex conjugate of 2i, which is -2i.
The polynomial can be written as:
(x - the first zero)(x - the second zero)(x - the third zero)(x - the fourth zero)
Using the given zeros, the polynomial is:
(x - (-1))(x - 3)(x - 2i)(x + 2i)
Simplifying further:
(x + 1)(x - 3)(x - 2i)(x + 2i)
Expanding the terms using FOIL (First, Outer, Inner, Last):
(x + 1)(x - 3) * (x - 2i)(x + 2i)
(x^2 - 2x + x - 3) * (x^2 + 2ix - 2ix - 4i^2)
(x^2 - x - 3) * (x^2 + 4)
Multiplying the two binomials:
x^2 * x^2 + x^2 * 4 + (-x) * x^2 + (-x) * 4 + (-3) * x^2 + (-3) * 4
x^4 + 4x^2 - x^3 - 4x - 3x^2 - 12
Combining like terms:
x^4 - x^3 + 4x^2 - 3x^2 - 4x - 12
Finally, the simplified form of the fourth-degree polynomial function is:
f(x) = x^4 - x^3 + x^2 - 4x - 12
To write a fourth-degree polynomial function with a leading coefficient of 1 given three of its zeros, including complex roots, we need to consider the Conjugate Root Theorem. The theorem states that if a polynomial function with real coefficients has a complex root \(a + bi\), where \(a\) and \(b\) are real numbers, then its conjugate, \(a - bi\), is also a root.
In this case, the given zeros are -1, 3, and 2i. Since 2i is a complex root, its conjugate, -2i, should also be a root. Therefore, we have four roots: -1, 3, 2i, and -2i.
To find the polynomial function, we can start by setting up the factors of the equation. Since -1 is a root, we have \((x + 1)\) as a factor. Similarly, since 3 is a root, we have \((x - 3)\) as a factor. Also, since 2i and -2i are roots, we have \((x - 2i)\) and \((x + 2i)\) as factors.
The next step is to multiply these factors together. We start with the (x + 1) and (x - 3) terms and then expand using the distributive property. Here's what it looks like:
\((x + 1)(x - 3)\)
\(= x^2 - 3x + x - 3\)
\(= x^2 - 2x - 3\)
Now, we multiply this expression by \((x - 2i)(x + 2i)\):
\((x^2 - 2x - 3)(x - 2i)(x + 2i)\)
\(= (x^2 - 2x - 3)(x^2 + 4)\)
Next, we multiply these two expressions using the distributive property:
\((x^2 - 2x - 3)(x^2 + 4)\)
\(= x^4 + 4x^2 - 2x^3 - 8x - 3x^2 - 12\)
Finally, combine like terms and write the polynomial function in simplified form:
\(f(x) = x^4 - 2x^3 + x^2 - 8x - 12\)