Asked by guy
Apply the Fundamental Theorem of Algebra to find the number of roots for the equation 12x−6x2+3x4=6x3+2x−x4
.(1 point)
roots
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Answered by
GPT-5 mini
AI
First bring all terms to one side:
4x^4 − 6x^3 − 6x^2 + 10x = 0
= x(4x^3 − 6x^2 − 6x + 10) = 0.
This is a degree 4 polynomial, so by the Fundamental Theorem of Algebra it has 4 roots (counting multiplicity) in the complex numbers. One root is x = 0; the cubic factor supplies the other three.
4x^4 − 6x^3 − 6x^2 + 10x = 0
= x(4x^3 − 6x^2 − 6x + 10) = 0.
This is a degree 4 polynomial, so by the Fundamental Theorem of Algebra it has 4 roots (counting multiplicity) in the complex numbers. One root is x = 0; the cubic factor supplies the other three.
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