Write a polynomial with rational coefficients in standard form satisfying the following conditions:

Question

Write a polynomial with rational coefficients in standard form satisfying the following conditions:

 

A)Degree 3 with only zeros x=-4 and x=2

 

B)Degree 3 with zeros x=3 and x=2i

 

C)Degree 3with zeros x=square root  -6 and x=-1

 

D)Degree 4with zeros x= square root 2 and x=i

Answer 1

If a is a zero of a polynomial it means that (x-a) is a factor.

So if x=2 is a zero than (x-2) is a factor.

If a polynomial has degree 3 it means the polynomial has 3 zeros.  If only two zeros are given then the third is a repeat of one of the given 2 zeros.

 

A y = (x-2)²(x+4) = (x²-4x+4)(x+4) = x³-4x²+4x+4x²-16x+16) = x³-12x+16 is one possible answer

the others can all be solved the same way; the only caveat is that when there is an imaginary root the conjugate is also a factor