find a polynomial P(x) with real coefficients having a degree 4, leading coefficient 3, and zeros 4-i and 4i

Question

Find a polynomial P(x) with real coefficients having a degree 4, leading coefficient 3, and zeros 4-i and 4i

Answer 1

Since P(x) has real coefficients, complex roots occur in conjugate pairs.  

 

So, since 4i is a root, so is -4i, and since 4-i is a root, so is 4+i.

 

Also, since P(x) has degree 4, it can't have more than 4 roots.  

 

P(x) = 3[x-4i][x-(-4i)][x-(4-i)][x-(4+i)]

 

      = 3(x²+16)[(x-4)+i][(x-4)-i]

 

      = 3(x²+16)[(x-4)²-i²]

 

      = 3(x²+16)(x²-8x+17) = 3(x⁴-8x³+17x²+16x²-128x+272)

 

                                      = 3(x⁴-8x³+33x²-128x+272)

 

                                      = 3x⁴-24x³+99x²-384x+816