(x-1)(x-3).jpeg "(x+1)(x-1)(x-3)")
Factoring and Solving the Expression (x+1)(x-1)(x-3)
The expression (x+1)(x-1)(x-3) is a factored polynomial. Let's explore its properties and how to solve it.
Understanding the Factored Form
The expression is already in factored form, meaning it's expressed as a product of simpler expressions. This form is helpful for:
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Finding the roots (or zeros): The roots are the values of x that make the expression equal to zero. Since the expression is a product, it equals zero when any of the factors equals zero. Therefore, the roots are:
- x + 1 = 0 => x = -1
- x - 1 = 0 => x = 1
- x - 3 = 0 => x = 3
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Graphing the function: The roots represent the points where the graph of the function y = (x+1)(x-1)(x-3) crosses the x-axis. The factored form also tells us the behavior of the function near the roots.
Expanding the Expression
We can expand the expression to get a polynomial in standard form:
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Start with the first two factors: (x+1)(x-1) = x² - 1 (This is a difference of squares pattern)
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Multiply the result by the third factor: (x² - 1)(x-3) = x³ - 3x² - x + 3
Therefore, the expanded form of the expression is x³ - 3x² - x + 3.
Applications
This expression can be used in various mathematical contexts, such as:
- Solving equations: Setting the expression equal to zero and solving for x would give us the same roots we found earlier.
- Finding the zeros of a polynomial function: The roots of the expression represent the x-intercepts of the function y = (x+1)(x-1)(x-3).
- Analyzing the behavior of the function: The factored form provides insights into the function's behavior around its roots.
Conclusion
The expression (x+1)(x-1)(x-3) is a factored polynomial with roots at x = -1, x = 1, and x = 3. Understanding the factored form allows us to easily find the roots and analyze the behavior of the corresponding function. It's important to remember that this expression can be expanded into a standard polynomial form, which can be useful in different applications.