Find the polynomial with real coefficients of the smallest possible

ikavumacj

ikavumacj

Answered question

2021-12-03

Find the polynomial with real coefficients of the smallest possible degree for which I and 1+I are zeros and in which the coefficient of the highest power is 1.

Answer & Explanation

Knes1997

Knes1997

Beginner2021-12-04Added 11 answers

To find the polynomial with real coefficients 
Assume that f(x) is a polynomial with real coefficients, with I and (1+i) as its zeros.Since the zeroes i and (1+i) are complex number and so its conjugates are also the zeroes of the polynomial f(x) with the real coefficient. 
So, the zeroes of the polynomial f(x) are -i,i, (1+i) and (1-i). 
Then, the polynomial f(x) will be of degree 4. 
If a is a zero of the polynomial f(x), then (x-a) is a factor of the polynomial f(x). 
So, 
f(x)=[x-(1+i)][x-(1-i)](x-i)(x+i) 
=[x1i][x1+i](x2i2) 
=[(x1)i][(x1)+i](x2+1) (Since, i2=1
=[(x1)2i2](x2+1) 
=[(x1)2+1](x2+1) 
=[x22x+1+1](x2+1) 
=[x22x+2](x2+1) 
=x42x3+2x2+x22x+2 
=x42x3+3x22x+2 
Hence, the required polynomial with the real coefficient of the smallest possible degree is 
f(x)=x42x3+3x22x+2

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?