(x^7-3)(x^7+7)-(x^7+2)^2

3 min read Jun 17, 2024

Simplifying the Expression: (x^7-3)(x^7+7)-(x^7+2)^2

This article will guide you through simplifying the expression (x^7-3)(x^7+7)-(x^7+2)^2. We'll break down each step to make it easy to understand.

Step 1: Expanding the Products

Let's start by expanding the products using the distributive property:

(x^7 - 3)(x^7 + 7):
This is a product of two binomials, which can be expanded as follows: (x^7 - 3)(x^7 + 7) = x^7 * x^7 + x^7 * 7 - 3 * x^7 - 3 * 7 = x^14 + 7x^7 - 3x^7 - 21 = x^14 + 4x^7 - 21
(x^7 + 2)^2: This is a square of a binomial, which can be expanded as follows: (x^7 + 2)^2 = (x^7 + 2)(x^7 + 2) = x^7 * x^7 + x^7 * 2 + 2 * x^7 + 2 * 2 = x^14 + 2x^7 + 2x^7 + 4 = x^14 + 4x^7 + 4

Step 2: Combining the Expanded Terms

Now, let's substitute the expanded terms back into the original expression:

(x^7-3)(x^7+7)-(x^7+2)^2 = (x^14 + 4x^7 - 21) - (x^14 + 4x^7 + 4)

Step 3: Simplifying the Expression

Finally, we can simplify the expression by removing the parentheses and combining like terms:

x^14 + 4x^7 - 21 - x^14 - 4x^7 - 4 = -21 - 4 = -25

Conclusion

Therefore, the simplified form of the expression (x^7-3)(x^7+7)-(x^7+2)^2 is -25. Notice that the terms involving x^14 and x^7 canceled out, leaving us with a constant value.