(x-3)(x-2)(x-4)%3D120.jpeg "(x-1)(x-3)(x-2)(x-4)=120")
Solving the Equation (x-1)(x-3)(x-2)(x-4) = 120
This equation represents a fourth-degree polynomial equation, and we can solve it by following these steps:
1. Expanding the Equation
First, we need to expand the left side of the equation. Let's multiply the terms step-by-step:
- Step 1: (x-1)(x-3) = x² - 4x + 3
- Step 2: (x-2)(x-4) = x² - 6x + 8
- Step 3: (x² - 4x + 3)(x² - 6x + 8) = x⁴ - 10x³ + 35x² - 50x + 24
Now our equation becomes: x⁴ - 10x³ + 35x² - 50x + 24 = 120
2. Rearranging the Equation
Subtract 120 from both sides to get a standard polynomial equation:
x⁴ - 10x³ + 35x² - 50x - 96 = 0
3. Finding the Roots (Solutions)
Solving this fourth-degree polynomial equation can be challenging. There are a few methods to approach this:
- Factoring: Try to factor the polynomial into simpler expressions. This might not be straightforward for this specific equation.
- Rational Root Theorem: This theorem helps you find potential rational roots. It states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term (in our case, -96) and q is a factor of the leading coefficient (in our case, 1).
- Numerical Methods: For equations that are difficult to factor, numerical methods like the Newton-Raphson method or bisection method can be used to find approximate solutions.
4. Checking the Solutions
Once you find potential roots using the above methods, substitute them back into the original equation to verify if they are true solutions.
Important Note:
The solution to this equation might involve complex numbers, as fourth-degree polynomials can have up to four roots (real or complex).
By employing these steps and appropriate techniques, you can find the solutions to the equation (x-1)(x-3)(x-2)(x-4) = 120.