-(x%5E2%2B4).jpeg "(x^4+8x^2+16) (x^2+4)")
Factoring and Simplifying (x^4 + 8x^2 + 16)(x^2 + 4)
This expression represents the product of two quadratic polynomials. Let's break down how to factor and simplify it.
Factoring the Expressions
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Recognize the pattern in (x^4 + 8x^2 + 16): This expression is a perfect square trinomial. It can be factored as (x^2 + 4)^2.
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Factor (x^2 + 4): While this expression itself doesn't factor further using real numbers, we can leave it as is.
Simplifying the Expression
Now that we've factored the expressions, we can simplify the entire product:
(x^4 + 8x^2 + 16)(x^2 + 4) = (x^2 + 4)^2 (x^2 + 4)
Therefore, the simplified form is (x^2 + 4)^3
Further Exploration
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Expanding the Expression: While the simplified form is concise, we can expand it further:
- (x^2 + 4)^3 = (x^2 + 4)(x^2 + 4)(x^2 + 4)
- This would require multiplying the terms repeatedly to get a polynomial of degree 6.
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Complex Numbers: It's worth noting that (x^2 + 4) can be factored if we allow complex numbers. The factors would be (x + 2i) and (x - 2i), where 'i' is the imaginary unit (√-1).
This example demonstrates how understanding factoring patterns and algebraic simplification can lead to concise representations of complex expressions.