%5E2.jpeg "(x^2+5)^2")
Expanding and Simplifying (x^2 + 5)^2
The expression (x^2 + 5)^2 represents the square of the binomial (x^2 + 5). To simplify this expression, we can use the FOIL method or the square of a binomial formula.
Using the FOIL Method
The FOIL method stands for First, Outer, Inner, Last. This helps us remember how to multiply two binomials.
- First: Multiply the first terms of each binomial: x^2 * x^2 = x^4
- Outer: Multiply the outer terms of the binomials: x^2 * 5 = 5x^2
- Inner: Multiply the inner terms of the binomials: 5 * x^2 = 5x^2
- Last: Multiply the last terms of each binomial: 5 * 5 = 25
Now, add all the terms together: x^4 + 5x^2 + 5x^2 + 25
Finally, combine like terms: x^4 + 10x^2 + 25
Using the Square of a Binomial Formula
The square of a binomial formula states: (a + b)^2 = a^2 + 2ab + b^2
In our case, a = x^2 and b = 5. Substituting these values into the formula:
(x^2 + 5)^2 = (x^2)^2 + 2(x^2)(5) + 5^2
Simplifying: x^4 + 10x^2 + 25
Conclusion
Both methods lead to the same simplified expression: x^4 + 10x^2 + 25. This is the expanded form of (x^2 + 5)^2.
Remember, using the square of a binomial formula can be faster and more efficient for simplifying expressions like this. However, understanding the FOIL method is crucial for expanding other types of binomial expressions.