Question

According to the rational root theorem, which of the following are possible zeros of P(x) = 4x^3 + 2x^2 + 6x + 6? List all correct answers.

a) -1
b) -2
c) -3
d) -6
e) All of the above

Answer 1

According to the Rational Root Theorem, all the given options -1, -2, -3, and -6 are possible zeros of the polynomial P(x) = 4x^3 + 2x^2 + 6x + 6.

Step-by-step explanation:

To determine possible rational zeros of the polynomial P(x) = 4x^3 + 2x^2 + 6x + 6 using the Rational Root Theorem, we look at factors of the constant term and the leading coefficient. For the constant term 6, its factors are ±1, ±2, ±3, and ±6. For the leading coefficient 4, its factors are ±1, ±2, and ±4. The possible rational zeros are therefore the divisors of the constant term divided by the divisors of the leading coefficient. These are:

±1/1, ±2/1, ±3/1, ±6/1, ±1/2, ±2/2, ±3/2, ±6/2, ±1/4, ±2/4, ±3/4, ±6/4

Which simplifies to:

• ±1
• ±2
• ±3
• ±6
• ±1/2
• ±3/2

All of the options a) -1, b) -2, c) -3, and d) -6 from the question are on our list of possible rational zeros, so the answer is e) All of the above.