Expand and simplify. (x+4)squared
The Correct answer and Explanation is:
To expand and simplify the expression (x+4)2(x + 4)^2(x+4)2, we will use the binomial theorem or the property of squaring a binomial.
Step 1: Understand the Binomial Expansion
The expression (a+b)2(a + b)^2(a+b)2 can be expanded using the formula:(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2
In our case, a=xa = xa=x and b=4b = 4b=4. Plugging in these values into the formula gives us:(x+4)2=x2+2(x)(4)+42(x + 4)^2 = x^2 + 2(x)(4) + 4^2(x+4)2=x2+2(x)(4)+42
Step 2: Perform the Calculations
Now, we calculate each term separately:
- a2=x2a^2 = x^2a2=x2
- 2ab=2(x)(4)=8x2ab = 2(x)(4) = 8x2ab=2(x)(4)=8x
- b2=42=16b^2 = 4^2 = 16b2=42=16
Step 3: Combine the Terms
Now, we combine all the terms we calculated:(x+4)2=x2+8x+16(x + 4)^2 = x^2 + 8x + 16(x+4)2=x2+8x+16
Step 4: Final Expression
Thus, the expanded and simplified form of (x+4)2(x + 4)^2(x+4)2 is:x2+8x+16x^2 + 8x + 16×2+8x+16
Explanation
The process of expanding a binomial involves applying the square of a sum formula, which emphasizes the importance of recognizing patterns in algebra. The first term in the expansion, x2x^2×2, represents the square of the variable itself. The middle term, 8x8x8x, is derived from the interaction between the two terms in the binomial. Specifically, it is twice the product of the first and second terms of the binomial (here, xxx and 444), which illustrates how changes in either term influence the outcome. The final term, 161616, is the square of the constant 444.
This method is not only useful for binomials but also serves as a foundation for understanding polynomial expansion, factorization, and algebraic manipulation in more complex expressions. Mastering these principles allows for greater flexibility in solving equations and performing calculus operations in higher mathematics. Hence, recognizing and practicing binomial expansions is crucial in developing a solid foundation in algebra.