Help please, I am completely lost

Question

What polynomial function has zeros 1, 1+i and 1-i

 

a) P(x)= x³-4x²+3x-1

b) P(x)= x³+2x²-3x-3

c) P(x)= x³-3x²+4x-2

d) P(x)= x³-x²+2x+1

 

Write the simplest polynomial function with the zeros 2, -i, sqt5, and -2 

 

a) P(x)= x⁵-2x⁴-8x³+20x²+15x-50=0

b) P(x)= x⁵-2x⁴-10x³+16x²+25x²-80x+100=0

c) P(x)= x⁵-2x⁴-8x³-20x²-65x-50=0

d) P(x)= x⁶-4x⁵-4x⁴+36x³-25x²-80x+100=0

Answer 1

Hi Alexandra,

 

For the first question, you can try plugging in 1 for x and see if you get zero in the equations because it is easy to test 1.

When you do this, only the polynomial in c will result in zero. Since it is multiple choice, you don't even need to find the complex roots.

 

However, the formal way to solve both questions will be to do synthetic or long division with the given real roots. If you get a zero remainder, it means that the number is a solution and you will end up with a lower degree polynomial. Once you reduce it to second degree, you can use the quadratic formula to find the complex roots.

 

Good luck!