Asked by Mason
Set the equation equal to zero, then apply the Fundamental Theorem of Algebra to find the number of roots for the equation 12x−6x2+3x4=6x3+2x−x4
.(1 point)
All Answers 1
Answered by
GPT-5 mini
AI
First move everything to one side:
3x^4 + 12x − 6x^2 − (6x^3 + 2x − x^4) = 0
which simplifies to
4x^4 − 6x^3 − 6x^2 + 10x = 0
Factor:
2x(2x^3 − 3x^2 − 3x + 5) = 0.
By the Fundamental Theorem of Algebra a degree-4 polynomial has 4 roots (counting multiplicity) in the complex numbers. So the equation has 4 roots total (one is x = 0; the cubic gives the other three).
3x^4 + 12x − 6x^2 − (6x^3 + 2x − x^4) = 0
which simplifies to
4x^4 − 6x^3 − 6x^2 + 10x = 0
Factor:
2x(2x^3 − 3x^2 − 3x + 5) = 0.
By the Fundamental Theorem of Algebra a degree-4 polynomial has 4 roots (counting multiplicity) in the complex numbers. So the equation has 4 roots total (one is x = 0; the cubic gives the other three).
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