Final answer:
To factor the polynomial function f(x) and solve the equation f(x) = 0, find the roots by factoring the polynomial using the constant term and the coefficient of the linear term. The factored form of the polynomial is (x - 3)(x + 7). Solving for x, the solutions are x = 3 and x = -7.
Step-by-step explanation:
To factor the polynomial function f(x) and solve the equation f(x) = 0, we need to find the roots of the polynomial. The given polynomial is f(x) = x² + 9x² + 11x - 21. To factor it, we can look for two numbers that multiply together to give the constant term (-21) and add up to give the coefficient of the linear term (11). The factors of -21 are -1, 1, -3, 3, -7, and 7. After testing all possible pairs, we find that -3 and 7 are the numbers that satisfy both conditions. So, the factored form of the polynomial is f(x) = (x - 3)(x + 7). Setting this expression equal to zero, we can solve for x by setting each factor equal to zero and solving for x. Therefore, the solutions of the equation are x = 3 and x = -7.