%5E4-16(x%2B5)%5E2.jpeg "(x+5)^4-16(x+5)^2")
Factoring (x+5)^4 - 16(x+5)^2
This expression can be factored by recognizing a pattern and using the difference of squares factorization. Here's how:
1. Identify the pattern:
Notice that the expression has the form of a squared term minus another squared term:
- (x+5)^4 is the square of (x+5)^2
- 16(x+5)^2 is the square of 4(x+5)
2. Apply the difference of squares:
The difference of squares factorization states: a² - b² = (a + b)(a - b)
Let's apply this to our expression:
- a = (x+5)^2
- b = 4(x+5)
Therefore, we can rewrite the expression as:
(x+5)^4 - 16(x+5)^2 = [(x+5)^2 + 4(x+5)][(x+5)^2 - 4(x+5)]
3. Simplify further:
We can factor out (x+5) from both terms in each bracket:
- [(x+5)^2 + 4(x+5)] = (x+5)(x+5 + 4) = (x+5)(x+9)
- [(x+5)^2 - 4(x+5)] = (x+5)(x+5 - 4) = (x+5)(x+1)
4. Final factored form:
Putting it all together, the completely factored expression is:
(x+5)^4 - 16(x+5)^2 = (x+5)(x+9)(x+5)(x+1) = (x+5)²(x+9)(x+1)
Conclusion:
By recognizing the pattern and applying the difference of squares factorization, we were able to simplify the expression (x+5)^4 - 16(x+5)^2 into its completely factored form: (x+5)²(x+9)(x+1).