DUE AT MIDNIGHT PLEASE HELP! THANK YOU

Question

Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 4- i, and passing through the origin.

Answer 1

A degree 5 polynomial needs 5 factors:

 

y = a(x+b)(x+c)(x+d)(x+e)(x+f)

• a = the leading coefficient
• b, c, d, e, and f are the roots or zeros of the polynomial

 

We need to find a, b, c, d, e, and f:

• a = the leading coefficient = 1 (given)
• b = i (given)
• If i is a root, so is -i, so c = -i
• d = 4-i  (given)
• If 4-i is a root, so is 4+i, so e = 4+i
• f = unknown at the moment

 

y = (x+i)(x-i)(x+4-i)(x+4+i)(x+f)

 

To find f, plug in the known point (0,0).

 

0 = (0+i)(0-i)(0+4-i)(0+4+i)(0+f)

0 = 18f

0 = f

 

Final equation is:

 

y = (x+i)(x-i)(x+4-i)(x+4+i)(x)

 

Multiply it out to put it into standard form.