Problem 10 - Elementary Algebra

Question

For the following problems, find the products. $$ (x+5)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of the expression \((x+5)^{2}\) is \(x^{2} + 10x + 25\).

Step 1: Use the distributive property

To find the product of \((x+5)^{2}\), we first need to expand the expression using the distributive property. This means that we are multiplying \((x+5)\) by itself: $$ (x+5)(x+5) $$

Step 2: Apply the FOIL method

To find the product of the expanded expression, we will apply the FOIL (First, Outer, Inner, Last) method. This method involves multiplying the first terms, the outer terms, the inner terms, and the last terms, and then adding the results together. First terms: \(x * x = x^{2}\) Outer terms: \(x * 5 = 5x\) Inner terms: \(5 * x = 5x\) Last terms: \(5 * 5 = 25\)

Step 3: Simplify the expression

Now that we've applied the FOIL method, let's add the results together to simplify the expression: $$ x^{2} + 5x + 5x + 25 $$ Combine like terms: $$ x^{2} + 10x + 25 $$ The product of the expression \((x+5)^{2}\) is \(x^{2} + 10x + 25\).

Key Concepts

Distributive Property

Mastering the distributive property is essential in algebra, especially when dealing with polynomials. This principle is the cornerstone for many operations, including the FOIL method. It allows us to multiply a single term by each term within a parenthesis individually. To put it simply, if you have an equation such as \(a(b + c)\), the distributive property lets you 'distribute' the multiplication of \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\).

When dealing with more complex situations, like squaring a binomial \((x+5)^2\), we first recognize that it is equivalent to \((x + 5)(x + 5)\). This is where we distribute each term of the first binomial over the second binomial. Consequently, the distributive property enables us to break down the problem into smaller, more manageable parts that can be solved step by step.

Simplifying Algebraic Expressions

After expanding algebraic expressions using the distributive property, simplification is the next critical step. Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form. Like terms are terms that have the same variable raised to the same power. For instance, in our example \(x^2 + 5x + 5x + 25\), the terms \(5x\) and \(5x\) are like terms because they both contain the variable \(x\) to the power of one.

To simplify, we add the coefficients of like terms, which in our exercise is adding together the coefficients of the \(5x\) terms, giving us \(10x\). The simplified form of the expression is \(x^2 + 10x + 25\), which is much easier to work with than the expanded form. In essence, simplifying helps make complex algebraic expressions more understandable and is a fundamental skill to building a strong foundation in algebra.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials together, a process often simplified by the FOIL method. A binomial is a polynomial with two terms, such as \((x+5)\). When we multiply one binomial by another, such as \((x+5)(x+5)\), we apply the FOIL method to ensure we account for each unique pair of multiplications.

The FOIL acronym stands for First, Outer, Inner, Last, representing the four products to be calculated:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of each binomial.
• Inner: Multiply the inner terms of each binomial.
• Last: Multiply the last terms of each binomial.

Once these multiplications are performed, the resulting terms are combined to produce the final result. Understanding this method allows students to handle a wide variety of multiplication problems involving polynomials efficiently and with confidence.