Question

use fundamental theorem algebra to find roots of 4x^-6x^3+2x^7=8x^5+3x^6

how many roots does it have

Answer 1

I’m not sure I parsed your polynomial correctly — the expression "4x^-6x^3+2x^7=8x^5+3x^6" is ambiguous. Could you confirm the intended exponents/plus/minus signs? For example, did you mean
- 4x − 6x^3 + 2x^7 = 8x^5 + 3x^6 ?
or something else?

General fact you can use (Fundamental Theorem of Algebra):
- Any nonzero polynomial of degree n (with complex coefficients) has exactly n complex roots counted with multiplicity.

So the steps are:
1. Move everything to one side to get a single polynomial P(x) = 0.
2. Determine the degree n of P(x).
3. By the Fundamental Theorem of Algebra, P(x) has exactly n complex roots (counting multiplicity). The number of real roots is ≤ n.

If your equation is the example 4x − 6x^3 + 2x^7 = 8x^5 + 3x^6, then
- rearranged: 2x^7 − 3x^6 − 8x^5 + 6x^3 − 4x = 0,
- that is a degree-7 polynomial, so it has exactly 7 complex roots (counting multiplicity). One obvious root is x = 0 (factor out x).

If you intended a different polynomial, paste it again exactly (use ^ for exponents) and I’ll give the specific count/factorization.