Form a polynomial, f(x) with real coefficients having the given degree and zeros. Degree: 4 ; Zeros: 4i and 5i

Really need help! don't know where to start.

Review the properties of polynomials with real coefficients. If they have complex roots, the roots appear in conjugate pairs. So, since you have roots 4i and 5i, the other roots are -4i and -5i.

Recall that x^2 + a^2 = (x-ai)(x+ai)

So, your polynomial is (x^2+16)(x^2+25) = x^4 + 41x^2 + 400

To form a polynomial with real coefficients having the given degree and zeros, we can use the complex conjugate property.

Since the zeros given are 4i and 5i, their complex conjugates would be -4i and -5i.

To find the polynomial, we need to use the fact that for a quadratic equation with complex zeros, the factors are (x - a)(x - b), where a and b are the complex conjugate pair of zeros.

Therefore, the factors for the given zeros would be (x - 4i)(x + 4i) and (x - 5i)(x + 5i).

Multiplying these factors together, we get:

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

Expanding this expression, we have:

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

(x^2 - 16i^2)(x^2 - 25i^2)

Simplifying, we have:

(x^2 + 16)(x^2 + 25)

Expanding this expression further, we get:

(x^2)(x^2) + (x^2)(25) + (16)(x^2) + (16)(25)

x^4 + 25x^2 + 16x^2 + 400

Finally, simplifying the expression, we get:

f(x) = x^4 + 41x^2 + 400

Therefore, the polynomial with real coefficients and degree 4, with zeros 4i and 5i, is f(x) = x^4 + 41x^2 + 400.

To form a polynomial with the given zeros, we can use the concept of complex conjugate zeros. Since 4i and 5i are zeros of the polynomial, their conjugates -4i and -5i will also be zeros.

The general form of a polynomial is given as:

f(x) = a(x - m1)(x - m2)(x - m3)(x - m4)

where a is a constant and m1, m2, m3, m4 are the zeros of the polynomial.

In this case, the zeros are 4i, 5i, -4i, and -5i. Putting these values into the general form, we get:

f(x) = a(x - 4i)(x - 5i)(x + 4i)(x + 5i)

Now, to simplify this expression, we can use the difference of squares identity, which states that (x - y)(x + y) = x^2 - y^2.

Applying this identity, we can rewrite the expression as:

f(x) = a((x^2 - (4i)^2)(x^2 - (5i)^2)

Now, simplify further:

f(x) = a(x^2 + 16)(x^2 + 25)

This is the polynomial of degree 4 with zeros at 4i and 5i.