Find the root of
x
3−2x2−5x+6=0
A. 1, 2, -3
B. -1, 2, 3
C. 1, 2, 3
D. 1, -2, 3
To find the roots of the equation x^3 - 2x^2 - 5x + 6 = 0, we can use synthetic division or factor it using the rational root theorem.
Using synthetic division, we can test each of the answer choices to see which one is a root. Let's try option C, which is 1, 2, and 3.
For x = 1:
1 | 1 - 2 - 5 + 6
1 - 1 - 6
- 6
The result is not zero, so 1 is not a root.
For x = 2:
2 | 1 - 2 - 5 + 6
2 0 - 10
- 4
The result is not zero, so 2 is not a root.
For x = 3:
3 | 1 - 2 - 5 + 6
3 3 - 6
0
The result is zero, so 3 is one root of the equation.
Therefore, the correct answer is D. 1, -2, 3.
To find the root of the equation x^3 - 2x^2 - 5x + 6 = 0, we can utilize a factoring technique. By evaluating the equation at various possible roots, we can find the values that satisfy the equation.
We start by trying the value x = 1:
(1)^3 - 2(1)^2 - 5(1) + 6 = 1 - 2 - 5 + 6 = 0
Since x = 1 is a root, we have already found one of the roots. To determine the other roots, we can divide the polynomial expression by (x - 1) using synthetic division:
1 │ 1 -2 -5 6
│ +1 -1 -6
──────────────
1 -1 -6 0
The result of the synthetic division is the quotient 1x^2 - 1x - 6 = 0, which we can further factor as (x - 2)(x + 3) = 0. This gives us the other roots as x = 2 and x = -3.
Hence, the roots of the equation x^3 - 2x^2 - 5x + 6 = 0 are 1, 2, and -3. Thus, the answer is option C: 1, 2, 3.