00:01
In this question, we are asked to find the roots of the even polynomial, given that 1 plus i is one with the root of this polynomial.
00:08
Here, we need to note that this is a polynomial with real coefficients, and all polynomials with real coefficients come whose roots are complex, come with complex conjugate roots.
00:25
So this means if 1 plus i is a 1 root of this polynomial, 1 minus i, which is a good, complex conjugate of 1 plus i also has to be a root of this polynomial.
00:44
This means that we can rewrite this polynomial as 1 plus i times, sorry, x minus 1 plus i times x minus i times x minus 1 minus i times some quadratic.
01:19
It's going to be a monic.
01:21
So x squared plus bx plus c.
01:28
The coefficient in front of x squared is 1 because x to the 4 has a coefficient 1.
01:37
Now this equals to, we can regroup the terms in the first two factors as x minus 1 minus i times x minus 1 plus i.
02:04
And this is a difference of squares in the first two factors.
02:08
And we are going to get x minus 1 squared minus i squared.
02:25
I call that i squared is negative 1 and x minus 1 squared expands as x squared minus 2x plus 1 and i squared is negative 1 and negative i squared is also plus 1.
02:45
So our polynomial factors as x squared minus 2x plus 2 times x squared plus bx plus c...