Pre calculus 12 grade

Question

Write the polynomial with real coefficients having the degree 4 and zeros (-2i,3,3)

Answer 1

4 degree

with 4 zeros 2i, 3, 3 and -2i as imaginary roots come in conjugate pairs

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

= (x^2+4)(x^2 -6x +9)

= x^4 -6x^3 +9x^2 +4x^2 -24x +36

= x^4 - 6x^3 +13x^2 - 24x +36

or any multiple of the above, such as double each term

2x^4 -12x^3 +26x^2 -48x +72

Answer 2

Complex zeroes come in pairs, so the other complex zero is the conjugate of -2i ie +2i (you switch the sign of the imaginary part).

So the four zeroes are -2i, +2i, 3, 3

So the polynomial factorises as a ( x-2i) (x+2i) (x-3) (x-3)

a is an arbitrary real constand which you can choose, choosing a=1 makes life easiest.

Then you have to multiply out the factorisation

(x-2i) (x + 2i) = x^2 +2ix -2ix -4i^2 = x^2 +4 (as i^2 = -1)

(x-3) (x-3) = x^2 -6x + 9

(x^2 + 4) (x^2 - 6x + 9)

x^2 -6x +9

x^2 x^4 -6x^3 +9x^2

4 + 4x^2 -24x + 36

Result : x^4 -6x^3 +13 x^2 -24x + 36