(x-3)(x%2B2)(x%2B4).jpeg "(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:
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Set each factor equal to zero:
- x - 1 = 0
- x - 3 = 0
- x + 2 = 0
- x + 4 = 0
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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.