(x%2B1)(x%2B3)(x%2B5).jpeg "(x-1)(x+1)(x+3)(x+5)")
Factoring and Expanding (x-1)(x+1)(x+3)(x+5)
This expression represents the product of four linear factors. Let's explore how to factor and expand it.
Factoring
The expression is already factored into its simplest form:
(x-1)(x+1)(x+3)(x+5)
Each factor represents a linear equation, and multiplying them gives us the original expression.
Expanding
To expand the expression, we need to multiply each of the factors together. We can do this systematically:
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Multiply the first two factors: (x-1)(x+1) = x² - 1 (using the difference of squares pattern)
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Multiply the last two factors: (x+3)(x+5) = x² + 8x + 15
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Multiply the results from step 1 and 2: (x² - 1)(x² + 8x + 15) = x⁴ + 8x³ + 15x² - x² - 8x - 15
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Combine like terms: x⁴ + 8x³ + 14x² - 8x - 15
Therefore, the expanded form of the expression is x⁴ + 8x³ + 14x² - 8x - 15.
Understanding the Result
This expansion represents a polynomial of degree 4. It has:
- Four roots: x = 1, x = -1, x = -3, and x = -5, which correspond to the zeros of each linear factor.
- Four turning points: This indicates a curve with multiple changes in direction.
Applications
Understanding how to factor and expand expressions like this is fundamental in algebra and calculus. It is used in:
- Solving equations: By setting the expression equal to zero, we can find the roots of the equation.
- Graphing functions: Knowing the roots and turning points helps visualize the graph of the function.
- Finding maxima and minima: By using calculus techniques, we can determine the maximum and minimum points of the function.
In summary, the expression (x-1)(x+1)(x+3)(x+5) can be factored and expanded, revealing its underlying structure and providing valuable information for solving equations and understanding the behavior of the corresponding function.