To solve the equation 4x^3 - 140x^2 + 1200x - 1872 = 0, we can use the Rational Root Theorem to find the possible rational roots.
The Rational Root Theorem states that if a polynomial has a rational root, then it will be a factor of the constant term divided by a factor of the leading coefficient. In this case, the constant term is -1872 and the leading coefficient is 4.
So, the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±31, ±48, ±62, ±93, ±124, ±186, ±248, ±372, ±624, ±936, and ±1872.
We can try these values for x and see if any of them satisfy the equation. By testing, we find that x = 8 is a root of the equation.
Now that we have found one root, we can use polynomial long division or synthetic division to divide the original polynomial by (x - 8) to get a quadratic equation. By dividing 4x^3 - 140x^2 + 1200x - 1872 by (x - 8), we get:
4x^3 - 140x^2 + 1200x - 1872
divided by (x - 8)
We get a quadratic equation x^2 - 24x + 234 = 0 after the first division.
Now, we can solve this quadratic equation by factoring or using the quadratic formula:
x^2 - 24x + 234 = 0
(x - 12)(x - 12) = 0
x = 12
Therefore, the roots of the equation 4x^3 - 140x^2 + 1200x - 1872 = 0 are x = 8, x = 12, and x = 12 (repeated root).
solve the equation 4x^3 - 140x^2 + 1200x - 1872 = 0
1 answer