What are the possible rational zeros of f(x) = x⁴ + 2x³ - 3x² - 4x - 12?

Solution:

We can use the rational zero theorem to find rational zeros of a polynomial.

By Rational Zero Theorem,

If P(x) is a polynomial with integer coefficients and if is a zero of P(x) (P( ) = 0),

then p is a factor of the constant term of P(x)

q is a factor of the leading coefficient of P(x)

Possible value of rational zero is p/q

Given, f(x) = x⁴ + 2x³ - 3x² - 4x + 12

Here, constant term, p = -12

Leading coefficient, q = +1

The factors of the constant term -12 are ±1, ±2, ±3, ±4, ±6, ±12.

The factor of the leading coefficient is ±1.

Possible values of rational zeros p/q = ±1/±1, ±2/±1, ±3/±1, ±4/±1, ±6/±1, ±12/±1

Therefore, the possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±12.

What are the possible rational zeros of f(x) = x⁴ + 2x³ - 3x² - 4x - 12?

Summary:

The possible rational zeros of f(x) = x⁴ + 2x³ - 3x² - 4x - 12 are ±1, ±2, ±3, ±4, ±6, ±12.