If  ф = -1+4√-3, then what is the value of the expression (`phi^4+7phi^3+59phi^2-5phi+12` )

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to raise `phi` to different powers to evaluate the expression, hence I suggest you to start raising `phi`  to square such that:

`phi^2 = (-1+4*sqrt(-3))^2`

You need to remember complex number theory that tells that `sqrt(-1)=i` , hence `sqrt(-3)=i*sqrt 3` .

Substituting `i*sqrt 3`  for `sqrt(-3)`  yields:

`phi^2 = (4i*sqrt 3 - 1)^2`

You need to use the formula for `(x-y)^2`  to evaluate `phi^2`  such that:

`(x-y)^2 = x^2 - 2xy + y^2`

Hence, `phi^2 = (4i*sqrt 3)^2 - 2*4i*sqrt 3*1 + (-1)^2`

`phi^2 = 48i^2 - 8isqrt3 + 1`

Considering `i^2 = -1`  yields:

`phi^2 = -47 - 8isqrt3`

`` You need to evaluate `phi^3 = phi^2*phi`

`phi^3 = -(47+8isqrt3)(4i*sqrt 3 - 1)`

`phi^3 = -188isqrt3 + 47+ 96 +8isqrt3`

`phi^3 = 143 - 180isqrt3`

Hence, `phi^4 = (phi^2)^2 = (47+8isqrt3)^2`

Using the formula `(x+y)^2 = x^2 + 2xy + y^2`  yields:

`(47+8isqrt3)^2 = 47^2 + 752isqrt3 - (8isqrt3)^2`

`(47+8isqrt3)^2 = 2209 + 752isqrt3 - 192`

`(47+8isqrt3)^2 = 2197 + 752isqrt3`

You may evaluate the expression substituting the values for `phi^2` , `phi^3` , `phi^4`  such that:

`phi^4 + 7phi^3 + 59phi^2 - 5phi + 12 = 2197 + 752isqrt3 + 7(143 - 180isqrt3) + 59(-47 - 8isqrt3) - 5(4i*sqrt 3 - 1) +12`

Opening the brackets yields:

`phi^4 + 7phi^3 + 59phi^2 - 5phi + 12 = 2197 + 752isqrt3 + 1001 - 1260isqrt3 - 2773 - 472isqrt3 - 20isqrt3 + 5 + 12`

`phi^4 + 7phi^3 + 59phi^2 - 5phi + 12 = 442 - 1000*sqrt3*i`

`phi^4 + 7phi^3 + 59phi^2 - 5phi + 12 = 442 - 1000*sqrt(-3)` Hence, evaluating the sum of powers of `phi ` yields

`phi^4 + 7phi^3 + 59phi^2 - 5phi + 12 = 442 - 1000*sqrt(-3)`