To find the zeros of the polynomial function f(x) = x³ + 3x² + 2x + 6, we need to solve the equation f(x) = 0.
This means we are looking for values of x that make the polynomial equal to zero. The polynomial can be factored or solved via various methods such as synthetic division, the Rational Root Theorem, or numerical approaches if necessary.
In this case, we can first check for possible rational roots using the Rational Root Theorem. The possible rational roots could be the factors of the constant term (6) divided by the leading coefficient (1). The factors of 6 are ±1, ±2, ±3, and ±6.
Testing these values, we see:
- f(1) = 1³ + 3(1)² + 2(1) + 6 = 12 (not a root)
- f(-1) = (-1)³ + 3(-1)² + 2(-1) + 6 = 6 (not a root)
- f(2) = 2³ + 3(2)² + 2(2) + 6 = 38 (not a root)
- f(-2) = (-2)³ + 3(-2)² + 2(-2) + 6 = 0 (root found)
We found that x = -2 is a zero of the function. We can then perform polynomial long division or synthetic division to factor f(x) by (x + 2).
After factoring it out, we find the other corresponding zeros by solving the resulting quadratic. The remaining polynomial factors down further, and any additional zeros can also be found using the quadratic formula if necessary.
Ultimately, the zeros of the equation give the x-values at which the polynomial intersects the x-axis. The complete solution provides insight into the roots and further behavior of the function, revealing all points where f(x) is equal to zero.