%2Fx%5E3%2B2x%5E2%2B4x%2B8.jpeg "(x^4-16)/x^3+2x^2+4x+8")
Factoring and Simplifying the Expression (x^4 - 16) / (x^3 + 2x^2 + 4x + 8)
This expression involves both factoring and simplifying rational expressions. Let's break it down step by step:
Factoring the Numerator
The numerator, (x^4 - 16), is a difference of squares. We can factor it as follows:
- (x^4 - 16) = (x^2 + 4)(x^2 - 4)
Further, the second factor (x^2 - 4) is also a difference of squares:
- (x^2 - 4) = (x + 2)(x - 2)
Therefore, the fully factored numerator is: (x^2 + 4)(x + 2)(x - 2)
Factoring the Denominator
The denominator, (x^3 + 2x^2 + 4x + 8), can be factored by grouping:
- (x^3 + 2x^2) + (4x + 8)
- x^2(x + 2) + 4(x + 2)
- (x^2 + 4)(x + 2)
Simplifying the Expression
Now we have the simplified expression:
[(x^2 + 4)(x + 2)(x - 2)] / [(x^2 + 4)(x + 2)]
We can cancel out the common factors (x^2 + 4) and (x + 2) leaving us with:
x - 2
Conclusion
The simplified form of the expression (x^4 - 16) / (x^3 + 2x^2 + 4x + 8) is x - 2. Remember that this simplification is valid as long as x ≠ -2 and x^2 ≠ -4.