Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 8, -14, and 3 + 9i f(x) = x4 - 11x3 + 72x2 - 606x + 10,080 f(x) = x4 - 303x2 + 1212x - 10,080 f(x) = x4 - 11x3 - 72x2 + 606x - 10,080 f(x) = x4 - 58x2 + 1212x - 10,080

Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.  8, -14, and 3 + 9i   f(x) = x4 - 11x3 + 72x2 - 606x + 10,080   f(x) = x4 - 303x2 + 1212x - 10,080   f(x) = x4 - 11x3 - 72x2 + 606x - 10,080   f(x) = x4 - 58x2 + 1212x - 10,080

tkhunny · Answer

What's your plan?

Zero of 8 means a factor of what?
Zero of -14 means a factor of what?
Zero of 3+9i means what other zero and a factor of what?

anonymous · Answer

i know its going to be (x-8)(x+14) but i dont know what to do with the 3-9i

anonymous · Answer

@tkhunny

tkhunny · Answer

The answer to that question is in this hint: "with real coefficients"

If 3+9i is a zero, then, just like the others (x - (3+9i)) is a factor.

However, if this is all we do, we are NOT going to get Real Coefficients.  We need to throw in a Complex Conjugate.  Thus, 3-9i must also be a zero and (x-(3-9i)) must also be a factor.  Multiply those two and you should get a nice quadratic trinomial with REAL coefficients.

This is the deal.  If we are to get Real Coefficients, Complex zeros MUST occur in Conjugate Pairs.  Simple as that.

Make sense?