Question

Form a polynomial, f(x), with real coefficients having the given degree and zeros.

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

Answer 1

To form a polynomial with real coefficients, given non-real zeros 4i and 5i, include their conjugates -4i and -5i. Multiply the corresponding factors to get f(x) = (x^2 + 16)(x^2 + 25), which simplifies to f(x) = x^4 + 41x^2 + 400.

Step-by-step explanation:

To form a polynomial, f(x), with real coefficients having a specified degree and given zeros, you must use the fact that non-real roots of polynomials with real coefficients come in conjugate pairs. Since 4i and 5i are given as zeros of the polynomial, their conjugates, -4i and -5i, must also be zeros of the polynomial.

The factors of the polynomial that correspond to these zeros are (x - 4i), (x + 4i), (x - 5i), and (x + 5i). To find the polynomial, these factors are multiplied together.

Multiplying the factors that come in conjugate pairs first:

Next, multiply these two results to obtain the polynomial:

(x^2 + 16)(x^2 + 25) = x^4 + 25x^2 + 16x^2 + 400 = x^4 + 41x^2 + 400

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

Answer 2

The question indicates that 4i and 5i are imaginary so, they cannot have real roots.

f(x) = (x - 4i)(x + 4i)(x - 5i)(x + 5i)
Then simplify it and it would look like this,

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

Simply it again.
x^4 + 16x^2 + 25x^2 + 400 x^4 + 41x^2 + 400
The polynomial that is required is
f(x) = x^4 + 41x^2 + 400