(x%5E2-9)(2x%2B5)%3D91.jpeg "(2x-7)(x^2-9)(2x+5)=91")
Solving the Equation: (2x-7)(x^2-9)(2x+5)=91
This problem involves solving a polynomial equation. Let's break down the steps to find the solutions for x:
1. Expand the Equation
First, we need to expand the equation by multiplying the factors:
- Expand (x^2-9): This is a difference of squares, which simplifies to (x+3)(x-3).
- Multiply the three factors: (2x-7)(x+3)(x-3)(2x+5) = 91
2. Simplify the Equation
- Multiply the first two factors: (2x^2 - x - 21)(x-3)(2x+5) = 91
- Multiply the result by (x-3): (2x^3 - 7x^2 - 34x + 63)(2x+5) = 91
- Finally, multiply by (2x+5): 4x^4 - 4x^3 - 113x^2 + 101x + 315 = 91
Now we have a fourth-degree polynomial equation: 4x^4 - 4x^3 - 113x^2 + 101x + 224 = 0
3. Finding Solutions
Unfortunately, there's no simple formula to directly solve fourth-degree equations. Here are common methods:
- Factoring: Try factoring the polynomial. It might be possible to factor it into smaller expressions, leading to solutions. However, this might be complex for a fourth-degree polynomial.
- Rational Root Theorem: This theorem helps identify potential rational roots of the polynomial. You can test these roots using synthetic division or by plugging them into the equation.
- Numerical Methods: For complex polynomials, numerical methods like the Newton-Raphson method or graphing calculators can provide approximate solutions.
4. Conclusion
Solving the equation (2x-7)(x^2-9)(2x+5)=91 involves expanding, simplifying, and then using appropriate methods to find the solutions for x. The exact solutions might require numerical methods or might involve complex numbers.