the rational root of the equation
2x³ - x² - 4x + 2 = 0 is
1/2
To find the rational roots of the equation 2x³ - x² - 4x + 2 = 0, we can use the Rational Root Theorem. According to the theorem, if a rational number p/q is a root of the equation, it must satisfy the conditions:
1. p is a factor of the constant term (in this case, 2).
2. q is a factor of the leading coefficient (in this case, 2).
First, let's list all the factors of the constant term (2): ±1, ±2.
Next, let's list all the factors of the leading coefficient (2): ±1, ±2.
Now, let's form all possible combinations of p/q using the factors we found.
Possible combinations:
±1/±1, ±1/±2, ±2/±1, ±2/±2
By simplifying these fractions, we get the potential rational roots:
±1, ±1/2, ±2
Now, substitute these values into the equation 2x³ - x² - 4x + 2 = 0 and check if they satisfy the equation.
For example, let's substitute x = 1 into the equation:
2(1)³ - (1)² - 4(1) + 2 = 0
2 - 1 - 4 + 2 = 0
-1 - 2 + 2 = 0
-1 = 0
Since the equation is not satisfied when x = 1, it is not a root. Repeat this process with all the potential rational roots until you find one or more values that satisfy the equation.
Please note that for this particular equation, there may or may not be any rational roots.