Problem 87 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$(2 x+5)(2 x-5)\left(4 x^{2}+25\right)$$

Step-by-Step Solution

Answer

The solution is: \(16x^4 - 625\).

Step 1: Distribute the First Two Terms

Distribute the first two terms, \(2x+5\) and \(2x-5\), using the distributive property (also known as the FOIL method for two binomials). FOIL stands for First, Outer, Inner, Last. Let's multiply:\n \[(2 x+5)(2 x-5)= 4x^2 - 25\]

Step 2: Distribute the Result with the Third Term

Now, distribute \(4x^2 - 25\) with the third term \(4x^2 + 25\). This multiplication will be simpler if treated as the multiplication of two binomials where the only difference is the sign.\n \[(4x^2 -25)(4x^2 + 25)=16x^4 - 625\]

Step 3: Conclusion

The multiplication of all three terms \((2 x+5)(2 x-5)(4 x^{2}+25)\) yields the solution \(16x^4 - 625\).