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List all the possible rational zeros of p(x) = 2x^3 - 3^2 + 4x - 6.
+-6, +-3, +-3/2, +-2, +-1, +-1/2
Only +3/2 is an actual root. Anonymous has correctly used the "p/q" theorem for a list of "possible" roots.
The other two actual roots are complex
To find the possible rational zeros of a polynomial, we can use the Rational Root Theorem.
The Rational Root Theorem states that if a rational number p/q is a zero of a polynomial equation with integer coefficients:
- The numerator p is a factor of the constant term (in this case, -6).
- The denominator q is a factor of the leading coefficient (in this case, 2).
For the given polynomial p(x) = 2x^3 - 3x^2 + 4x - 6, the constant term is -6 and the leading coefficient is 2. So, we are looking for rational zeros of the form p/q, where p is a factor of -6 and q is a factor of 2.
The factors of -6 are:
1, -1, 2, -2, 3, -3, 6, -6
The factors of 2 are:
1, -1, 2, -2
So, the possible rational zeros are the combinations of these factors:
±1, ±2, ±3, ±6
Hence, the possible rational zeros of p(x) = 2x^3 - 3x^2 + 4x - 6 are ±1, ±2, ±3, and ±6.