(x%2B2)(x%2B3)(x%2B8)%2B56.jpeg "(x-3)(x+2)(x+3)(x+8)+56")
Factoring and Solving the Expression (x-3)(x+2)(x+3)(x+8) + 56
This article explores the process of factoring and solving the expression (x-3)(x+2)(x+3)(x+8) + 56.
Understanding the Expression
The expression (x-3)(x+2)(x+3)(x+8) + 56 is a polynomial expression. It's a product of four linear factors and a constant term.
Factoring the Expression
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Recognizing the Pattern: Notice that the first four terms form a pattern. We can rewrite them as a product of two quadratic expressions:
(x-3)(x+3)(x+2)(x+8) = [(x-3)(x+3)][(x+2)(x+8)] = (x² - 9)(x² + 10x + 16)
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Expanding the Quadratics: Expand the quadratics to obtain a new expression:
(x² - 9)(x² + 10x + 16) = x⁴ + 10x³ + 16x² - 9x² - 90x - 144
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Simplifying: Combine like terms:
x⁴ + 10x³ + 7x² - 90x - 144
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Adding the Constant Term: Add the constant term from the original expression:
x⁴ + 10x³ + 7x² - 90x - 144 + 56 = x⁴ + 10x³ + 7x² - 90x - 88
Solving the Equation
Now, we have the simplified expression: x⁴ + 10x³ + 7x² - 90x - 88. To solve this equation, we need to find the values of x that make the expression equal to zero. This can be done through several methods, such as:
- Factoring: We could try to factor the simplified polynomial further. However, in this case, it's not easily factorable.
- Rational Root Theorem: This theorem helps identify potential rational roots, but it doesn't guarantee finding them.
- Numerical Methods: Methods like the Newton-Raphson method can approximate the roots to a desired degree of accuracy.
- Graphing: We can plot the function and observe where it crosses the x-axis (where the function is equal to zero).
Conclusion
Factoring and solving the expression (x-3)(x+2)(x+3)(x+8) + 56 involves recognizing patterns, simplifying the expression, and applying various techniques to find the solutions. While factoring the expression fully might be challenging, numerical methods and graphing can provide valuable insights into the roots of the equation.