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root equation .
- Thread starter ringokid
- Start date
BigGlenntheHeavy
Senior Member
- Joined
- Mar 8, 2009
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\(\displaystyle Descartes' \ Rule \ of \ Signs \ gives:\)
\(\displaystyle f(x) \ = \ 8x^5+77x^4+2x^3+8x^2-93x+42 \ \implies \ 2 \ changes\)
\(\displaystyle f(-x) \ = \ -8x^5+77x^4-2x^3+8x^2+93x+42 \ \implies \ 3 \ changes\)
\(\displaystyle Hence, \ we \ have \ 4 \ possibilities, \ to \ wit:\)
\(\displaystyle 2^+,3^-, 0^I \ or \ \ 2^+,1^-,2^I \ or \ \ 0^+,3^-,2^I, \ or \ 0^+,1^-,4^I \\)
\(\displaystyle Now, \ if \ we \ assume \ that \ the \ equations \ has \ 3 \ real \ roots, \ then,\)
\(\displaystyle only \ 2 \ possibilities \ exist, \ viz., \ 2^+,1^-,2^I \ or \ 0^+,3^-.2^I\)
\(\displaystyle Now, \ taking \ out \ my \ trusty \ TI-89, \ I \ get \ the \ following \ roots:\)
\(\displaystyle (.62347538,0),(.62348262,0), \ and \ (-9.623475,0) \ which \ coincides \ with \ 2^+,1^-,2^I\)
\(\displaystyle Note; \ You \ could \ also \ find \ the \ roots \ by \ the \ method \ of \ exhaustion \ or \ Newton's\)
\(\displaystyle Method \ if \ you \ prefer \ a \ lot \ of \ tedious \ grunt \ work.\)
\(\displaystyle f(x) \ = \ 8x^5+77x^4+2x^3+8x^2-93x+42 \ \implies \ 2 \ changes\)
\(\displaystyle f(-x) \ = \ -8x^5+77x^4-2x^3+8x^2+93x+42 \ \implies \ 3 \ changes\)
\(\displaystyle Hence, \ we \ have \ 4 \ possibilities, \ to \ wit:\)
\(\displaystyle 2^+,3^-, 0^I \ or \ \ 2^+,1^-,2^I \ or \ \ 0^+,3^-,2^I, \ or \ 0^+,1^-,4^I \\)
\(\displaystyle Now, \ if \ we \ assume \ that \ the \ equations \ has \ 3 \ real \ roots, \ then,\)
\(\displaystyle only \ 2 \ possibilities \ exist, \ viz., \ 2^+,1^-,2^I \ or \ 0^+,3^-.2^I\)
\(\displaystyle Now, \ taking \ out \ my \ trusty \ TI-89, \ I \ get \ the \ following \ roots:\)
\(\displaystyle (.62347538,0),(.62348262,0), \ and \ (-9.623475,0) \ which \ coincides \ with \ 2^+,1^-,2^I\)
\(\displaystyle Note; \ You \ could \ also \ find \ the \ roots \ by \ the \ method \ of \ exhaustion \ or \ Newton's\)
\(\displaystyle Method \ if \ you \ prefer \ a \ lot \ of \ tedious \ grunt \ work.\)
If we factor:
\(\displaystyle (x^{2}+9x-6)(8x^{3}+5x^{2}+5x-7)\)
The quadratic part discriminant, \(\displaystyle b^{2}-4ac=81-4(1)(-6)=105\), is > 0. Therefore, it has two real roots.
2 down.
The cubic part discriminant, \(\displaystyle b^{2}c^{2}-4c^{3}-4b^{3}d-27d^{2}+18bcd=(5)^{2}(5)^{2}-4(5)^{3}-4(5)(-7)-27(-7)^{2}+18(5)(5)(-7)=-4208\)
Since the discriminant < 0, then it has 1 real and two complex roots.
Thus, there is 1 real from the quadratic and two real from the cubic...making three real roots. The other two are complex.
\(\displaystyle (x^{2}+9x-6)(8x^{3}+5x^{2}+5x-7)\)
The quadratic part discriminant, \(\displaystyle b^{2}-4ac=81-4(1)(-6)=105\), is > 0. Therefore, it has two real roots.
2 down.
The cubic part discriminant, \(\displaystyle b^{2}c^{2}-4c^{3}-4b^{3}d-27d^{2}+18bcd=(5)^{2}(5)^{2}-4(5)^{3}-4(5)(-7)-27(-7)^{2}+18(5)(5)(-7)=-4208\)
Since the discriminant < 0, then it has 1 real and two complex roots.
Thus, there is 1 real from the quadratic and two real from the cubic...making three real roots. The other two are complex.