x² + x³ = 810
Rearrange in descending order:
x³ + x² - 810 = 0
If it has any rational solutions, they will be ± factors
of 810, which are:
±1, ±2, ±3, ±5, ±6, ±9, ±10, ±15, ±18, ±27, ±30, ±45,
±54, ±81, ±90, ±135, ±162, ±270, ±405, ±810
It has one sign change so it has 1 positive solution.
So we'll try the positive values with synthetic division.
Notice that we have to put in a place holder for there
is no x term, so we consider the equation as
x³ + x² + 0x - 810 = 0
We try 1:
1|1 1 0 -810
| 1 2 2
1 2 2 -808 No
We try 2:
2|1 1 0 -810
| 2 6 12
1 3 6 -798 No
We try 3:
3|1 1 0 -810
| 3 12 36
1 4 12 -774 No
We try 5:
5|1 1 0 -810
| 5 30 150
1 6 30 -660 No
We try 6:
6|1 1 0 -810
| 6 42 252
1 7 42 -558 No
We try 9:
9|1 1 0 -810
| 9 90 810
1 10 90 0 YES!
So x³ + x² - 810 = 0 factors as
(x - 9)(x² + 10x + 90) = 0
We use the zero-factor principle:
x - 9 = 0; x² + 10x + 90
x = 9; x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
Three solutions, one real and two imaginary:
9,
, and
Edwin