Consider the polynomial
f(x) = 2x^3 – 3x^2 – 8x – 3.
(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.
the rational zero theorem states that all rational zeroes of the given polynomial, the numerators have the form ±(factors of constant term) and the denominators have the form ±(factors of the coefficient of the highest term).
See for example:
http://mathworld.wolfram.com/RationalZeroTheorem.html
In the given case,
constant term = -3,
coeff. of highest term = 2
Possible rational zeroes are therefore:
1, 3, 1/2, 3/2, -1, -3, -1/2, -3/2
I will leave it to you to factorize the given polynomial to get the actual zeroes.
You are welcome to post your answer for verification.
(a) To find the possible rational zeros of a polynomial, we need to consider the factors of the constant term (c) and the factors of the leading coefficient (a) of the polynomial.
For the given polynomial f(x) = 2x^3 - 3x^2 - 8x - 3, the constant term is -3 and the leading coefficient is 2.
To find the factors of the constant term -3, we can check all combinations of integers that divide evenly into -3. In this case, the factors of -3 are ±1 and ±3.
To find the factors of the leading coefficient 2, we can check all combinations of integers that divide evenly into 2. In this case, the factors of 2 are ±1 and ±2.
So, the possible rational zeros of the given polynomial are all the possible combinations of factors of the constant term divided by factors of the leading coefficient. Therefore, the possible rational zeros are ±1/2, ±1, ±3/2, ±3.
(b) To find the zeros of the polynomial, we use the possible rational zeros we found in part (a) and check which ones are actually zeros by using synthetic division or long division.
To begin, let's test x = 1 as a possible zero using synthetic division:
1 | 2 -3 -8 -3
| 2 -1 -9
-------------
2 -1 -9 -12
Since the remainder is not zero, x = 1 is not a zero of the polynomial.
Next, let's test x = -1 as a possible zero using synthetic division:
-1 | 2 -3 -8 -3
| -2 5 3
------------
2 -5 -3 0
Since the remainder is zero, x = -1 is a zero of the polynomial.
Now, we have reduced the polynomial by one degree: f(x) = 2x^2 - 5x - 3.
Next, let's test x = -3/2 as a possible zero using synthetic division:
-3/2 | 2 -5 -3
| -3/2 0
------------
2 -4 -3
Since the remainder is not zero, x = -3/2 is not a zero of the polynomial.
Lastly, let's test x = 3 as a possible zero using synthetic division:
3 | 2 -4 -3
| 6 6
----------
2 2 3
Since the remainder is not zero, x = 3 is not a zero of the polynomial.
Hence, the zeros of the given polynomial f(x) = 2x^3 - 3x^2 - 8x - 3 are x = -1.