(x-3)(x%2B5)(x%2B7)%3D297.jpeg "(x-1)(x-3)(x+5)(x+7)=297")
Solving the Equation (x-1)(x-3)(x+5)(x+7) = 297
This equation presents a challenge as it involves a product of four linear factors. Here's how we can solve it:
1. Expanding the Equation
The first step is to expand the product of the factors. This can be done in a few ways:
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Pairwise Multiplication: Group the factors and multiply them in pairs:
- [(x-1)(x-3)][(x+5)(x+7)]
- This leads to (x² - 4x + 3)(x² + 12x + 35)
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Direct Expansion: Multiply the terms one by one, being careful with the signs.
After expansion, we'll get a fourth-degree polynomial equation.
2. Simplifying the Equation
Once the expansion is done, simplify the polynomial equation by combining like terms. This will give us a standard form:
ax⁴ + bx³ + cx² + dx + e = 297
3. Transforming to a Standard Form
Subtract 297 from both sides to get a standard fourth-degree polynomial equation:
ax⁴ + bx³ + cx² + dx + (e - 297) = 0
4. Finding Roots (Solutions)
Finding the roots of a fourth-degree polynomial equation can be complex. There are several methods:
- Factoring: Try to factor the polynomial into simpler expressions. This might not always be possible.
- Rational Root Theorem: This theorem helps find potential rational roots. It involves checking factors of the constant term (e - 297) and the leading coefficient (a).
- Numerical Methods: Methods like the Newton-Raphson method or graphical approaches can approximate the roots.
5. Solutions
The solutions to the equation (x-1)(x-3)(x+5)(x+7) = 297 represent the values of x that make the equation true. These solutions are called the roots of the equation.
Important Note: Solving a fourth-degree polynomial equation can be challenging. There might not be simple algebraic solutions, and numerical methods might be required to find approximate roots.