How do I solve this problem?

Question

im having a bit of trouble finishing my Precalc homework. The problem is “find a polynomial function with zeros of (-2, 4, 3+i (imaginary number)) I’m really struggling and it’s due tomorrow. Please help

Answer 1

Hello Seth,

First of all because you have an imaginary zero of 3 + i you also have the conjugate imaginary of 3 - i. So the first thing we need to do is put each answer in the form of a zero equation. Therefore, x=-2, x=4, x=3+i, and x=3-i. Now we can think of this as reversing solving polynomial equations, so put them back in factor form by moving the constants back to the left. That gives us (x+2)(x-4)(x-3-i)(x-3+i)=0. I always start with the imaginary factors, multiply the imaginary factors together to get x²-3x+ix-3x+9-3i-ix+3i-i². I did that by distributing. Now cancel out terms and collect like terms. Remember that i² is -1. You should get x²-6x+10. Remember that answer because you're going to use it again. Now multiply the (x+2)(x-4) together to get x²-2x-8. At this point you have (x²-6x+10)(x²-2x-8)=0 so we can now multiply the trinomials together to get x⁴-8x³+14x²+20x-80=0 and now you can change the 0 into f(x) to get the function. I hope this has helped!