To find the complex roots of the given polynomial, use the Rational Root Theorem to find a rational root. Then use polynomial long division to factor out that root. Finally, solve the resulting quadratic equation to find the remaining complex roots. Therefore, the complex roots of the given polynomial are x = -7, x = -18 + √330, and x = -18 - √330.
The given polynomial is f(x) = x² + x³ + 43x² + 49x - 294.
To find the complex roots of the polynomial, we can use the Rational Root Theorem to narrow down the possible solutions. The Rational Root Theorem states that if a polynomial equation has a rational root (p/q), then p must be a factor of the constant term (294) and q must be a factor of the leading coefficient (1).
In this case, the possible rational roots are ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42, ±49, ±98, ±147, and ±294. By testing these values, we find that x = -7 is a root of the polynomial. Therefore, (x + 7) is a factor of the polynomial.
Using polynomial long division, we can divide f(x) by (x + 7) to get the quotient f(x) = (x + 7)(x² + 36x - 42).
Now we can solve the quadratic equation x² + 36x - 42 = 0 to find the remaining roots. By using the quadratic formula, we get x = -18 ± √330.
Therefore, the complex roots of the given polynomial are x = -7, x = -18 + √330, and x = -18 - √330.