To find the possible rational zeros of a polynomial, we can use the Rational Zero Theorem. According to the theorem, if a polynomial has a rational zero, it must be of the form p/q, where p is a factor of the constant term (in this case, -2) and q is a factor of the leading coefficient (in this case, 3).
(a) Using the Rational Zero Theorem, the possible rational zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2 are the ratios of the factors of -2 (constant term) and 3 (leading coefficient).
The factors of -2 are: ±1, ±2
The factors of 3 are: ±1, ±3
So, the possible rational zeros are:
±(1/1), ±(1/3), ±(2/1), ±(2/3)
Simplifying these rational numbers gives us:
±1, ±1/3, ±2, ±2/3
(b) To find all the zeros of the polynomial, we can use these possible rational zeros and perform synthetic division or check them by substituting in the polynomial equation.
Let's try the possible zeros one by one. Start with the first possible zero, which is 1:
When we substitute x = 1 into the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2 and perform the calculations, we get:
f(1) = 3(1)^3 – 2(1)^2 – 7(1) – 2
= 3 - 2 - 7 - 2
= -8
Since f(1) is not equal to zero, 1 is not a zero of the polynomial.
We repeat this process for all the possible zeros until we either find the zeros or exhaust all possibilities.
After checking all the possible rational zeros, we find that the polynomial does not have any rational zeros.
To find the remaining zeros, we can use numerical methods or graph the polynomial to estimate the locations of the zeros. In this case, the remaining zeros are irrational and need to be approximated using numerical methods like Newton's method or a graphing calculator.