find a polynomial of least degree(having real coefficients) with zeros: 5, -2, 2i

multiply this out:

(x-5) (x^2+4)

I suspect you copied the problem wrong, it should be zeroes at 5, -2i, 2i

Unless there is the implicit assumption that if 2i is a root, so is -2i, so the polynomial would be

(x-5)(x+2)(x^2+4) = x^4 - 3x^3 - 6x^2 - 12x - 40

To find a polynomial with the given zeros, we need to consider the complex zeros as well. Since the polynomial has real coefficients, the complex zeros will occur in conjugate pairs. Therefore, if 2i is a zero, then its conjugate, -2i, must also be a zero.

To find the polynomial, we can use the zero product property. This property states that if a polynomial has a zero at a certain value, then the polynomial can be factored by (x - a), where "a" is the zero.

The zeros given are 5, -2, 2i, and -2i. Therefore, the polynomial can be factored as follows:

(x - 5)(x + 2)(x - 2i)(x + 2i)

Next, we can use the fact that (x - 2i)(x + 2i) = (x^2 + 4). So, the polynomial can be simplified as:

(x - 5)(x + 2)(x^2 + 4)

Now, we can multiply the factors together to find the polynomial:

(x - 5)(x + 2)(x^2 + 4)
= (x^2 + 4)(x - 5)(x + 2)
= (x^2 + 4)(x^2 - 3x - 10)
= x^4 - 3x^3 - 10x^2 + 4x^2 - 12x - 40
= x^4 - 3x^3 - 6x^2 - 12x - 40

Therefore, a polynomial with the given zeros is:
f(x) = x^4 - 3x^3 - 6x^2 - 12x - 40

To find a polynomial with real coefficients and given zeros, we need to use the concept of complex conjugate pairs. Since 2i is a zero, its conjugate is -2i (because complex roots always come in conjugate pairs).

To find the polynomial, we can write it in factored form using the given zeros:

(x - 5)(x - (-2))(x - 2i)(x - (-2i))

Simplifying this expression yields:

(x - 5)(x + 2)(x - 2i)(x + 2i)

Expanding the equation, we obtain:

(x - 5)(x + 2)(x^2 - (2i)^2)

(x - 5)(x + 2)(x^2 - 4i^2)

(x - 5)(x + 2)(x^2 - 4(-1))

(x - 5)(x + 2)(x^2 + 4)

Now we can distribute and simplify further:

(x^2 - 5x + 2x - 10)(x^2 + 4)

(x^2 - 3x - 10)(x^2 + 4)

Expanding one last time, we get:

x^4 + 4x^2 - 3x^3 - 10x^2 - 10x + 12x + 40

Rearranging the terms, we get:

x^4 - 3x^3 - 6x^2 + 2x + 40

Thus, the polynomial of least degree with real coefficients and zeros at 5, -2, and 2i is x^4 - 3x^3 - 6x^2 + 2x + 40.