Hi, can anyone show me how to do this problem? 100 points for this. in advance

z^2 + (-1 + 5·i)·z + 14 - 7·i = 0

1+2i  is a root

z= 1+2i

z^2 = (1+2i) (1+2i)

       = 1 +2i+2i +4i^2

       = 1 +4i -4

       = -3+4i

= (-1+5i) (1+2i)

-1+5i-2i+10i^2

  -1+3i-10

  -11+3i                

z^2 + (-1 + 5·i)·z + 14 - 7·i = 0

-3+4i  + -11+3i +14 - 7i

Combine like terms

-3-11 +14 +4i +3i-7i

0

So 1+2i is a root

1-2i  is not a root

z= 1-2i

z^2 = (1-2i) (1-2i)

       = 1 -2i-2i +4i^2

       = 1 -4i -4

       = -3-4i

= (-1+5i) (1-2i)

-1+5i+2i-10i^2

  -1+7i+10

  9+7i                

z^2 + (-1 + 5·i)·z + 14 - 7·i = 0

-3-4i  + 9+7i +14 - 7i

Combine like terms

-3+9 +14 -4i +7i-7i

20 -4i

So 1-2i is not a root

The complex conjugate being roots is only true for real coefficients

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step-by-step explanation:

the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

the root or roots of a polynomial are the values of the variable that makes the polynomial equal to 0. we can find the roots of a polynomial making the expression equal to 0 and then, isolating the variable (it it has just one variable).

for example, given the following polynomial:

the root or zero of the polynomial, can be found by making the expression equal to 0, and then, isolating the variable, so, finding the root we have:

therefore, we have that the root of the polynomial is equal to 2.

we must remember that the number of roots that a polynomial can have, will depend on the degree of the polynomial, a first degree polynomial have will have one single root, a second degree polynomial (quadratic expression) may have two roots and so.

hence the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

have a nice day!