list all of the possible rational zeros of f(x) = 3x^5 - 7x^3 + 2x - 15.
Let the rational roots (if any) be represented by p/q, where p, q ∈ℤ, p≠0, q≠0, and where p|-15 (constant term) and q|3 (leading coefficient).
Note: a|b means a divides b.
The possible candidates for rational roots are therefore:
±{1,3,5}/±{1,3}
or
±1, ±3, ±5, ±1/3, and ±5/3.
Evaluating f(x) for each of the candidates confirm that there are no rational roots.
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To find the real zeroes, we need to find the critical points (where f'(x) = 0) and evaluate f(x) at these points.
At all four critical points f'(x)<0, therefore there is only one real zero between 1.14 and infinity.
The single real (irrational) zero is at x=1.747 approx.
To find all the possible rational zeros of a polynomial function, you can use the Rational Root Theorem. According to this theorem, any rational root of the polynomial function f(x) = 3x^5 - 7x^3 + 2x - 15 can be represented as a fraction p/q, where p is a factor of the constant term (-15) and q is a factor of the leading coefficient (3).
Step 1: Find the factors of the constant term (-15). The factors of -15 are: ±1, ±3, ±5, ±15.
Step 2: Find the factors of the leading coefficient (3). The factors of 3 are: ±1, ±3.
Step 3: Combine the factors obtained from step 1 and step 2 to obtain all possible rational zeros.
The possible rational zeros of f(x) = 3x^5 - 7x^3 + 2x - 15 are:
±1/1, ±3/1, ±5/1, ±15/1, ±1/3, ±3/3 (which simplifies to ±1).
Therefore, the possible rational zeros are: ±1, ±3, ±5, ±15.
To find the possible rational zeros of a polynomial, we can use the Rational Root Theorem. According to the theorem, the possible rational zeros of a polynomial are of the form p/q, where p is a factor of the constant term (in this case -15) and q is a factor of the leading coefficient (in this case 3).
Now let's find the factors of -15: ±1, ±3, ±5, ±15.
And let's find the factors of 3: ±1, ±3.
Combining the factors, the possible rational zeros of f(x) = 3x^5 - 7x^3 + 2x - 15 are:
±1/1, ±1/3, ±3/1, ±3/3, ±5/1, ±5/3, ±15/1, ±15/3.
Simplifying these fractions, we get:
±1, ±1/3, ±3, ±1/3, ±5, ±5/3, ±15, ±5.
So, the possible rational zeros are:
±1, ±1/3, ±3, ±5, ±15, and ±5/3.