Asked by blake
apply the fundamental theorem of algebra to find the number of roots for the polynomial g\left(x\right)=7\:^{x6+2x-5}
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Answered by
GPT-5 mini
AI
I assume you mean the polynomial g(x) = 7x^6 + 2x − 5 (if you meant 7^(x^6+2x−5) tell me — that is not a polynomial).
By the Fundamental Theorem of Algebra, a nonzero polynomial of degree n has exactly n complex roots counted with multiplicity. Here deg g = 6, so g(x) has exactly 6 complex roots (counting multiplicity).
Additional notes:
- The coefficients are real, so nonreal roots occur in conjugate pairs.
- Since g(0) = −5 and g(x) → +∞ as x → ±∞, the Intermediate Value Theorem shows there is at least one positive real root and at least one negative real root — so at least 2 of the 6 roots are real. The remaining roots may be real or come in complex-conjugate pairs.
By the Fundamental Theorem of Algebra, a nonzero polynomial of degree n has exactly n complex roots counted with multiplicity. Here deg g = 6, so g(x) has exactly 6 complex roots (counting multiplicity).
Additional notes:
- The coefficients are real, so nonreal roots occur in conjugate pairs.
- Since g(0) = −5 and g(x) → +∞ as x → ±∞, the Intermediate Value Theorem shows there is at least one positive real root and at least one negative real root — so at least 2 of the 6 roots are real. The remaining roots may be real or come in complex-conjugate pairs.
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