(x-1)(x-3)(x+2)(x+4)

3 min read Jun 17, 2024
(x-1)(x-3)(x+2)(x+4)

Factoring and Solving the Equation (x-1)(x-3)(x+2)(x+4) = 0

This expression represents a polynomial equation in factored form. Let's break down what it means and how to solve it.

Understanding the Factored Form

The equation (x-1)(x-3)(x+2)(x+4) = 0 is already factored. This means it's written as a product of several simpler expressions, each representing a factor.

Key Principle: For a product of factors to equal zero, at least one of the factors must be zero.

Finding the Solutions

To solve the equation, we apply this principle:

  1. Set each factor equal to zero:

    • x - 1 = 0
    • x - 3 = 0
    • x + 2 = 0
    • x + 4 = 0
  2. Solve for x in each equation:

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

Therefore, the solutions to the equation (x-1)(x-3)(x+2)(x+4) = 0 are x = 1, x = 3, x = -2, and x = -4.

Interpreting the Solutions

These solutions represent the roots or zeros of the polynomial. They are the x-values where the polynomial function crosses the x-axis.

Graphically, these solutions correspond to the points where the graph of the polynomial intersects the x-axis.

Expanding the Expression

If we were to expand the original expression, we'd get a polynomial of degree 4. This is because we have four factors, each containing an 'x' term. The expanded form would be:

x⁴ + 2x³ - 11x² - 24x + 24

Note: Although the factored form is much easier to work with for solving, the expanded form is useful for other purposes like finding the y-intercept of the graph or understanding the end behavior of the function.

Summary

The equation (x-1)(x-3)(x+2)(x+4) = 0 is a factored polynomial equation. By setting each factor to zero, we find the solutions, which are x = 1, x = 3, x = -2, and x = -4. These solutions represent the roots or zeros of the polynomial, corresponding to the points where the graph intersects the x-axis.

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