(x%2B9)(x%2B5)%5E2%2B63.jpeg "(x+1)(x+9)(x+5)^2+63")
Factoring and Solving the Expression (x+1)(x+9)(x+5)^2 + 63
This article explores the process of factoring and solving the expression (x+1)(x+9)(x+5)^2 + 63.
Understanding the Expression
The given expression is a polynomial with a degree of 5, meaning it has five possible roots. It's a combination of linear and squared factors, making it potentially challenging to factor directly.
Strategies for Factoring
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Expansion and Grouping:
- Expand the expression to obtain a standard polynomial form.
- Attempt to group terms and factor by grouping. However, this approach can be quite tedious and may not lead to a successful factorization.
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Substitution:
- Consider a substitution to simplify the expression.
- Let y = (x+5).
- The expression becomes (y-4)(y+4)y^2 + 63, which simplifies to y^4 - 16y^2 + 63.
- This new expression is a quadratic in y^2. We can factor it as (y^2 - 7)(y^2 - 9).
- Substitute back y = (x+5) to get ((x+5)^2 - 7)((x+5)^2 - 9).
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Recognizing Patterns:
- Observe that the expression resembles the expansion of a perfect square.
- Notice that ((x+5)^2 - 7)((x+5)^2 - 9) can be rewritten as ((x+5)^2 - 8)^2 - 1.
- This pattern allows us to factor the expression further using the difference of squares formula: (a^2 - b^2) = (a+b)(a-b).
- The final factored expression becomes ((x+5)^2 - 8 + 1)((x+5)^2 - 8 - 1), which simplifies to ((x+5)^2 - 7)((x+5)^2 - 9).
Solving the Equation
To find the roots (solutions) of the expression, we set it equal to zero:
((x+5)^2 - 7)((x+5)^2 - 9) = 0
This equation is satisfied when either factor equals zero:
- (x+5)^2 - 7 = 0
- (x+5)^2 - 9 = 0
Solving for x in each case:
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(x+5)^2 = 7
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x + 5 = ±√7
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x = -5 ±√7
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(x+5)^2 = 9
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x + 5 = ±√9
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x = -5 ±3
Therefore, the solutions to the equation are:
- x = -5 + √7
- x = -5 - √7
- x = -2
- x = -8
Conclusion
Factoring and solving the expression (x+1)(x+9)(x+5)^2 + 63 involves recognizing patterns, using substitution, and applying factoring techniques. The solutions, or roots, of the equation are x = -5 + √7, x = -5 - √7, x = -2, and x = -8. This demonstrates the power of algebraic manipulations and pattern recognition in simplifying complex expressions.