Question

Find a rational zero of the polynomial function: f(x) = x³ + 4x² - x - 4.

Answer 1

To find a rational zero of the polynomial function f(x) = x³ + 4x² - x - 4, we can use the Rational Root Theorem. The possible rational zeros are ±1, ±2, ±4. By substituting these values into the polynomial, we find that the rational zeros are x = 1, x = -1, x = 2, and x = -2.

Step-by-step explanation:

To find a rational zero of the polynomial function f(x) = x³ + 4x² - x - 4, we can use the Rational Root Theorem. According to the theorem, any rational zero of the polynomial must be of the form p/q, where p is a factor of the constant term (-4) and q is a factor of the leading coefficient (1). So, the possible rational zeros are ±1, ±2, ±4. We can try these values by substituting them into the polynomial and checking if it equals zero:

1. Substituting x = 1: f(1) = 1³+4(1)²-1-4 = 0. Therefore, x = 1 is a rational zero.
2. Substituting x = -1: f(-1) = (-1)³+4(-1)²+1-4 = 0. Therefore, x = -1 is a rational zero.
3. Substituting x = 2: f(2) = 2³+4(2)²-2-4 = 0. Therefore, x = 2 is a rational zero.
4. Substituting x = -2: f(-2) = (-2)³+4(-2)²+2-4 = 0. Therefore, x = -2 is a rational zero.

So, the rational zeros of the polynomial function f(x) = x³ + 4x² - x - 4 are x = 1, x = -1, x = 2, and x = -2.