Find all possible roots for the polynomial \(3x^3 - 2x^2 - 5x + 10\)
Answer
Step 4: Substitute each possible root into the polynomial equation to see if it equals 0. After testing, we find that only \(-2\) and \(\frac{5}{3}\) are actual roots of the polynomial.
Step 1 ：Step 1: Apply the Rational Root Theorem, which states that any rational root, \(\frac{p}{q}\), of a polynomial equation \(ax^n + bx^{n-1} + . . . + k = 0\) can be expressed in the form \(\frac{p}{q}\), where p is a factor of the constant term (k) and q is a factor of the leading coefficient (a).
Step 2 ：Step 2: For the polynomial \(3x^3 - 2x^2 - 5x + 10\), the constant term is 10 and the leading coefficient is 3. The factors of 10 are \(\pm1, \pm2, \pm5, \pm10\) and the factors of 3 are \(\pm1, \pm3\).
Step 3 ：Step 3: Form all possible fractions \(\frac{p}{q}\) where p is a factor of 10 and q is a factor of 3. The possible rational roots are \(\pm1, \pm2, \pm5, \pm10, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{5}{3}, \pm\frac{10}{3}\).
Step 4 ：Step 4: Substitute each possible root into the polynomial equation to see if it equals 0. After testing, we find that only \(-2\) and \(\frac{5}{3}\) are actual roots of the polynomial.
