Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2.
(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.
(b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
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To find the possible rational zeros of a polynomial, we can use the Rational Zero Theorem. According to this theorem, any rational zero of a polynomial can be written in 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) To find all possible rational zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2, we need to determine the factors of the constant term (-2) and the leading coefficient (3).
The factors of -2 are ±1 and ±2, and the factors of 3 are ±1 and ±3. Now we can write down all the possible rational zeros using the combinations of these factors:
±1/1, ±1/3, ±2/1, ±2/3
These are all the possible rational zeros of the given polynomial.
(b) To find the zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2, we can use a process called synthetic division to evaluate each of the possible rational zeros and check for zeros.
Let's start with the first possible rational zero, x = 1. We perform synthetic division as follows:
1 | 3 -2 -7 -2
-------------------
| 3 1 -6 -8
The remainder is -8, which means x = 1 is not a zero of the polynomial.
Next, let's try x = -1:
-1 | 3 -2 -7 -2
-------------------
|-3 5 2 0
We got a remainder of 0, which means x = -1 is a zero of the polynomial.
Now we have factored f(x) as (x + 1) times a quadratic equation: f(x) = (x + 1)(3x^2 + 5x + 2).
To find the remaining zeros, we solve the quadratic equation 3x^2 + 5x + 2 = 0. This can be factored as (3x + 2)(x + 1) = 0, which means the remaining zeros are:
x = -1 (from the linear factor) and x = -2/3 (from the quadratic factor).
Therefore, all the zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2 are x = -1, x = -2/3, and the possible rational zeros we listed earlier: x = ±1/1 and x = ±2/3.