Problem 11 - Elementary Algebra

Question

For the following problems, find the products. $$ (x+8)^{2} $$

Step-by-Step Solution

Answer

Answer: x^2 + 16x + 64

Step 1: Identify a and b

In the given expression, the binomial is \((x+8)\), so we have \(a=x\) and \(b=8\).

Step 2: Apply the square of a binomial formula

Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), we'll replace \(a\) with \(x\) and \(b\) with \(8\): $$ (x+8)^2 = x^2 + 2(x)(8) + 8^2 $$

Step 3: Simplify the expression

Now, we'll calculate and simplify the expression: $$ x^2 + 16x + 64 $$ The product of the given expression \((x+8)^2\) is \(x^2 + 16x + 64\).

Key Concepts

Binomial Expansion

Understanding binomial expansion is essential when working with algebraic expressions that involve powers of binomials. In our example, the binomial is \(x+8\). Expanding this using binomial theorem would usually involve the Pascal's triangle or the binomial coefficients, especially for higher powers. However, when we're dealing with squares, it becomes much simpler.

The binomial expansion for the square of a binomial, which is \( (a+b)^2 \), always follows a specific pattern: it results in \( a^2 + 2ab + b^2 \). This pattern is derived from the distributive property of multiplication over addition, applied twice because of the exponent of 2. Knowing this pattern allows us to expand binomials quickly without resorting to lengthy multiplication.

Let's apply this to our case. The given binomial is raised to the second power, so we can use the pattern to expand \( (x+8)^2 \) directly to \( x^2 + 16x + 64 \) without going through the multiplication manually.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and involve numbers, variables, and arithmetic operations. In the given problem, \( (x+8)^2 \), the expression includes the variable \( x \) and the number 8. Understanding how to manipulate these expressions is key to simplifying and solving them.

An algebraic expression doesn't involve an equality sign, unlike an equation, which means it doesn't have a definitive solution but can be simplified or manipulated in various ways. In the context of our example, expanding and simplifying the expression are the main tasks at hand. Building your skills in working with algebraic expressions will allow you to tackle more complex problems in algebra with confidence.

Simplifying Expressions

Simplifying expressions is a process of reducing an algebraic expression to its simplest form. This means combining like terms, performing arithmetic operations, and minimizing the number of terms, if possible. For instance, in the expression we are considering, \( (x+8)^2 \), simplifying the expression involves applying the binomial expansion formula to eliminate the parentheses and then combining terms.

Simplifying makes expressions more understandable and easier to work with, especially when they are part of larger equations or when you are required to graph them. Armed with the knowledge of the square of a binomial pattern, simplification becomes much more straightforward, allowing for quick and error-free simplification.

Exponentiation

Exponentiation is an arithmetic operation that involves raising a number or expression to a power, indicating how many times to multiply the number or expression by itself. The expression \( (x+8)^2 \) represents the exponentiation of the binomial \( (x+8) \) to the power of 2. It translates to multiplying \( (x+8) \) by itself.

In algebra, understanding the laws of exponents is crucial for manipulating expressions with powers, as it helps in simplifying and solving more complex problems. The exponent of 2 in our example simplifies the process, but being familiar with exponentiation is indispensable when dealing with variables raised to higher, variable, or even negative exponents.