Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 4, -8, and 2 + 5i

Answer 1

The polynomial with given zeros 4, -8, and 2 + 5i is P(x) = x^4 - 4x^3 + 29x^2 + 32x - 232.

Step-by-step explanation:

To find a polynomial with given zeros, we can start by creating factors that correspond to these zeros. A polynomial that has 4, -8, and 2 + 5i as zeros will require a conjugate pair for the complex zero, which means that 2 - 5i is also a zero (since coefficients are real).

The factors will correspond to the zeros as follows: (x - 4) from the zero 4, (x + 8) from the zero -8, and (x - (2 + 5i)) and (x - (2 - 5i)) from the zero 2 + 5i and its conjugate 2 - 5i.

Expanding these factors out, we get:

(x - 4) for the zero 4

(x + 8) for the zero -8

(x - 2 - 5i)(x - 2 + 5i) for the zeros 2 + 5i and 2 - 5i

The last two factors can be multiplied as follows:

(x - 2 - 5i)(x - 2 + 5i) = (x - 2)^2 - (5i)^2 = x^2 - 4x + 4 + 25 = x^2 - 4x + 29

Next, we multiply all three factors to obtain the polynomial:

P(x) = (x - 4)(x + 8)(x^2 - 4x + 29)

Multiplying these, the polynomial in standard form should be:

P(x) = x^4 - 4x^3 + 29x^2 + 32x - 232

Answer 2

Hello,

(x-4)(x+8)(x-2)(x-5i)(x+5i)
=(x^2+4x-32)(x-2)(x^2+25)
=(x^3+2x^2-40x+64)(x^2+25)
=x^5 +2x^4 -15x^3 +114 x^2-1000x+1600