Question

Find all rational zeros of the polynomial. (enter your answers as a comma-separated list. enter all answers including repetitions.) p(x) = x5 − 4x4 − 8x3 42x2 − 9x − 54 x = incorrect: your answer is incorrect. write the polynomial in factored form.

a)−3,−2,2,3
b) −5,−4,4,5
c) −6,−3,3,6
d) −8,−6,6,8

Answer 1

The rational zeros of the polynomial are -3, -2, 2, and 3.

Step-by-step explanation:

The polynomial is given as p(x) = x^5 - 4x^4 - 8x^3 + 42x^2 - 9x - 54. To find the rational zeros, we can use the Rational Root Theorem. According to the theorem, the rational zeros are of the form p/q, where p is a factor of the constant term (-54) and q is a factor of the leading coefficient (1). These factors can be positive or negative. In this case, the possible rational zeros are ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54.

We can now test each of these values by substituting them into the polynomial equation. By using synthetic division or polynomial long division, we can check if each value is a zero. The zeros are the values that give a remainder of zero. After testing, we find that the rational zeros are x = -3, -2, 2, and 3. However, -3 and -2 are repeated zeros, so the final answer is x = -3, -2, 2, 3.