find the possible roots for:
3x^3-6x^2+11x-6=0
I assume you mean possible rational roots. If a fraction p/q is a root,
p divides 6 and
q divides 3
so, find the factors of 3 and 6 and list them out
To find the roots of the equation 3x^3 - 6x^2 + 11x - 6 = 0, we can make use of the Rational Root Theorem and synthetic division. Let's go step by step to find the possible roots:
Step 1: Identify the coefficients of the polynomial.
The polynomial in question is 3x^3 - 6x^2 + 11x - 6 = 0. The coefficients are:
a = 3
b = -6
c = 11
d = -6
Step 2: Determine the possible rational roots.
Using the Rational Root Theorem, we know that the possible rational roots are given by the following formula:
p/q, where p is a divisor of the constant term (d = -6) and q is a divisor of the leading coefficient (a = 3).
The divisors of 3 are ±1 and ±3, and the divisors of 6 are ±1, ±2, ±3, and ±6. So, the possible rational roots are:
±1/1, ±1/3, ±2/1, ±2/3, ±3/1, or ±3/3.
Simplifying, the possible rational roots are:
±1, ±1/3, ±2, ±2/3, ±3, or ±1.
Step 3: Test each possible root using synthetic division.
Using synthetic division, we can check each possible root to determine if it is a factor of the polynomial. A factor will yield a remainder of zero.
Let's take each possible root and substitute it into the synthetic division:
For x = -1:
-1 │ 3 -6 11 -6
──── 3 -9 -2
3 -9 -2
For x = -1/3:
-1/3 │ 3 -6 11 -6
───── -3 3
3 -9 -3
For x = 2:
2 │ 3 -6 11 -6
─── 12 12
3 6 6
For x = 2/3:
2/3 │ 3 -6 11 -6
────── 2 0
3 0 -6
Based on the synthetic division results, we see that x = -1 and x = 2/3 are the roots that make the remainder zero, indicating that they are the solutions to the equation 3x^3 - 6x^2 + 11x - 6 = 0.
Therefore, the possible roots for the equation 3x^3 - 6x^2 + 11x - 6 = 0 are x = -1 and x = 2/3.