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Finding Real and Complex Roots of a 6th Degree Polynomial

The 7th degree polynomial x^7 - 3x^6 - 7x^4 + 21x^3 - 8x + 24 has a factor (x - 3)

(a) Divide x^7 - 3x^6 - 7x^4 + 21x^3 - 8x + 24 by x - 3 and thus:

(b) Express it in the form (x - 3)(ax^6 + bx^3 + c)

(c) By putting Z = x^3, find all the factors, real or complex of the 6th degree polynomial and thus:

(d) Express x^7 - 3x^6 - 7x^4 + 21x^3 - 8x + 24 as the product of seven linear factors.

Solution Preview

The 7th degree polynomial x^7 - 3x^6 - 7x^4 + 21x^3 - 8x + 24 has a factor (x - 3)

(a) Divide x^7 - 3x^6 - 7x^4 + 21x^3 - 8x + 24 by x - 3 and thus:
x-3) x^7 - 3x^6 - 7x^4 + 21x^3 - 8x + 24 (x^6-7x^3-8
x^7 - 3x^6
- 7x^4 + 21x^3
- 7x^4 + 21x^3
- 8x + 24
- 8x + 24
...

Solution Summary

Real and imaginary roots of a polynomial are found. The solution is detailed and well presented.

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