Express z^4 + z^3 + z^2 + z + 1 as a product of two real quadratic factors.    this is a complex numbers question

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thilina-g | College Teacher | (Level 1) Educator

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The given expression is,


If we want to write this as a product of two quadratic expressions, the following equation satisfies our requirement.

`z^4+z^3+z^2+z+1 = (z^2+az+1)(z^2+bz+1)`

a and b are constants which we have to determine later.

Expanding the RHS,

`z^4+z^3+z^2+z+1 = z^4+(a+b)z^3+(1+ab+1)z^2+(a+b)z+1`

Now we can compare the coefficients on either side to find values for a and b.

Through coefficient of `z^3` terms, we get,

`a+b = 1`      -----------> Eq:1

Through coefficient of `z^2` terms, we get,

`1+ab+1 = 1`

`ab = -1` ------------------> Eq:2

Similarly through coefficient of `z ` terms, we get,

`a+b = 1`    This is same as equation 1, so we can ignore this.


Thus, we have two equations,

`a+b = 1` and `ab = -1`

Substitutig for b in Eq:1 from Eq:2

`b = -1/a`


By simplyfing this we get another quadratic equation as below.

`a^2-a-1 = 0`

By solving this we can find values for a.

`a = (1+-sqrt(5))/2`

Now there are two answers for a, they are `(1+sqrt(5))/2` and `(1-sqrt(5))/2` . For each of these values we can find respective value for b.

For `a = (1+sqrt(5))/2`

`b = -1/((1+sqrt(5))/2) = -2/(1+sqrt(5))`

By multiplying both numerator and denominator by `(1-sqrt(5))` ,

`b = (-2(1-sqrt(5)))/((1+sqrt(5))(1-sqrt(5)))`

`b = (-2(1-sqrt(5)))/(1-5)`

`b = 2/4(1-sqrt(5))`

`b = (1-sqrt(5))/2`


When `a = (1+sqrt(5))/2` , `b is (1-sqrt(5))/2` .

If you find the value of b for `a = (1-sqrt(5))/2` , you would get,

`b = (1+sqrt(5))/2` .

Therefore we have only one set of answers for a and b, because of the symmetry involved in the answers.

So we will select `a = (1+sqrt(5))/2` and `b = (1-sqrt(5))/2`


Therefore the two factors of the given expression are,

`z^4+z^3+z^2+z+1 = (z^2+(1+sqrt(5))/2z+1)(z^2+(1-sqrt(5))/2z+1)`