Solve for x: x^19 - 2x^18 + 7x^17 - 12x^6 + 6x^15 = 0

Question

Solve for x: x^19 − 2x^18 + 7x^17 − 12x^6 + 6x^15 = 0

 

Please help and thank you!
Full answer please!

Answer 1

Assuming that x⁶ should be x¹⁶, we have:

 

x¹⁵(x⁴ - 2x³ + 7x² - 12x + 6)

 

1 is a root of the second factor.  Divide the second factor synthetically by x - 1:

 

1⌋     1     -2     7     -12     6

                  1    -1       6     -6

        1      -1     6      -6      0

 

So, the original polynomial = x¹⁵(x-1)(x³-x²+6x-6)

 

1 is a root of x³-x²+6x-6.  Divide synthetically by x-1:

 

1⌋     1     -1      6      -6

                  1      0       6

        1       0      6       0

 

The original polynomial = x¹⁵(x-1)²(x² + 6)

 

Setting the polynomial equal to zero, we get x = 0, 1, √(6)i, -√(6)i