write a polynomial function of least degree with integral coefficients whose zeros include 4 and 2i
If (x - 2i ) then also the complex conjugate (x+2i) must be a factor. Then of course (x-4) so:
(x-2i)(x+2i)(x-4)
=(x^2+4)(x-4)
=x^3 - 4 x^2 + 4 x -16
To find a polynomial function with integral coefficients that has the given zeros, we can use the concept of complex conjugates. Since 2i is a zero, its complex conjugate -2i must also be a zero.
Now, we can write the polynomial function as follows:
(x - 4)(x - 2i)(x + 2i)
Multiplying this out, we get:
(x - 4)(x^2 + 4)
Expanding further:
x(x^2 + 4) - 4(x^2 + 4)
Applying the distributive property:
x^3 + 4x - 4x^2 - 16
Rearranging and combining like terms, the final polynomial function is:
f(x) = x^3 - 4x^2 + 4x - 16
To write a polynomial function of least degree with integral coefficients whose zeros include 4 and 2i, we can use the fact that complex conjugate pairs of numbers are zeros of polynomials with real coefficients.
Since 2i is a zero, its conjugate -2i will also be a zero of the polynomial. Hence, the zeros of the polynomial are 4, 2i, and -2i.
To create a polynomial with these zeros, we can start by setting up the factors:
(x - 4)(x - 2i)(x + 2i)
Since we want the polynomial to have integral coefficients, we need to get rid of the imaginary terms. To do this, we can multiply out the complex conjugate factors:
(x - 4)(x - 2i)(x + 2i) = (x - 4)(x^2 - (2i)^2)
= (x - 4)(x^2 + 4)
Now, we can simplify further:
(x - 4)(x^2 + 4) = x^3 + 4x - 4x^2 - 16
= x^3 - 4x^2 + 4x - 16
Thus, the polynomial function of least degree with integral coefficients whose zeros include 4 and 2i is:
f(x) = x^3 - 4x^2 + 4x - 16.