Math Homework help

Question

Give an expression for  using only real-valued coefficients, having degree 4, and roots  , , and .

Answer 1

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

 

So, since 2-5i is a root, so is 2+5i

 

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

P(x) = (x + 5)(x - (2 - 5i))(x - (2 + 5i))(x-7)

P(x)= x^4-6x^3+2x^2+82x-1015

Answer 2

If c is a root, then x-c is a factor. Also, if a+bi is a complex root, then so is a-bi.

f(x) = (x+5)(x-7)[(x-(2-5i))(x-(2+5i))]

= (x+5)(x-7)[((x-2)+5i)((x-2)-5i)] = (x+5)(x-7)(x²-4x+29)