(x-a)(x-b)(x-c)(x-d) Formula

3 min read Jun 17, 2024
(x-a)(x-b)(x-c)(x-d) Formula

Understanding the (x-a)(x-b)(x-c)(x-d) Formula

The formula (x-a)(x-b)(x-c)(x-d) represents a polynomial equation where a, b, c, and d are distinct constants. This formula is particularly useful for finding the roots of a polynomial equation, which are the values of x that make the equation equal to zero.

Understanding the Roots

When a polynomial is factored into the form (x-a)(x-b)(x-c)(x-d), each factor corresponds to a root of the polynomial. To find the roots, we set each factor equal to zero and solve for x:

  • (x-a) = 0 => x = a
  • (x-b) = 0 => x = b
  • (x-c) = 0 => x = c
  • (x-d) = 0 => x = d

Therefore, the roots of the polynomial equation (x-a)(x-b)(x-c)(x-d) = 0 are a, b, c, and d.

Applications

This formula has wide applications in various fields, including:

  • Algebra: Finding the roots of polynomial equations is crucial in solving algebraic problems, especially in quadratic equations.
  • Calculus: The formula is used to find the zeros of functions and to analyze their behavior.
  • Physics: The formula is applied in solving problems related to motion, energy, and waves.
  • Engineering: This formula helps in designing structures, analyzing circuits, and solving problems in various engineering disciplines.

Example

Let's consider the polynomial equation (x-2)(x+1)(x-3)(x-4) = 0. Using the formula, we can easily identify the roots:

  • x = 2
  • x = -1
  • x = 3
  • x = 4

Therefore, the roots of the polynomial equation are 2, -1, 3, and 4.

Conclusion

The (x-a)(x-b)(x-c)(x-d) formula is a powerful tool for finding the roots of polynomial equations. It provides a simple and efficient method for understanding the behavior of polynomial functions and has wide applications across various fields.

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