List all the possible rational zeros of
f(x)=3x^3-x^2+2x=2.
Please help me. I don't understand how to do this. Can someone show me please.
If a polynomial f(x)=0 has rational roots, the numerator must be a factor of the constant term, and the denominator must be a factor of the highest-power coefficient.
so, since you have f(x) = 3x^3-x^2+2x+2
any rational roots must have a numerator which divides 2, and a denominator which divides 3. So, any rational root must be in the list
±1/1, ±2/1, ±1/3, ±2/3
This comes from the fact that if you have (in this case) 3 roots (p,q,r) for
f(x) = ax^3 + bx^2 + cx + d
pqr = -d/a
pq + pr + qr = c/a
p + q + r = -b/a
To find the possible rational zeros of a polynomial function, you can use the Rational Root Theorem. The Rational Root Theorem states that if a polynomial has a rational root (a zero) in the form p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p is a divisor of the constant term and q is a divisor of the leading coefficient.
First, let's rewrite the equation in standard form by moving all the terms to one side:
f(x) = 3x^3 - x^2 + 2x - 2 = 0
The constant term is -2, and the leading coefficient is 3. We need to find the divisors of 3 and -2.
The divisors of 3 are 1 and 3, while the divisors of -2 are 1 and 2 (since we're looking for rational zeros, we consider both positive and negative divisors).
Now, we construct all possible combinations of the divisors. These combinations represent the possible rational zeros of the function.
For the positive combinations:
1/1, 1/3, 2/1, 2/3
For the negative combinations:
-1/1, -1/3, -2/1, -2/3
Simplifying these fractions further (if possible), we get:
1, 1/3, 2, 2/3, -1, -1/3, -2, -2/3
Therefore, the possible rational zeros of the function f(x) = 3x^3 - x^2 + 2x - 2 = 0 are 1, 1/3, 2, 2/3, -1, -1/3, -2, and -2/3.