Problem 69 - Algebra 1: Concepts and Skills

Question

Find the product. \((x+10)(x+10)\)

Step-by-Step Solution

Answer

The product of the expression \((x+10)(x+10)\) is \(x^2 + 20x + 100\)

Step 1: Applying the FOIL method

To multiply these two binomials together, we can utilize the FOIL method. That is, multiply the first terms, then outer terms, then the inner terms, and finally the last terms. Since both binomials are the same, we can square each term individually. Therefore, \( (x+10)(x+10) = x^2 + 2*10x + 10^2 \)

Step 2: Simplifying the expression

Once we've multiplied the terms according to the FOIL method, it is important to simplify the resulting expression. In our case, it simplifies to \( x^2 + 20x + 100 \)

Key Concepts

Binomial Multiplication

The process of binomial multiplication involves taking two binomial expressions and producing a single polynomial. For instance, consider the expression textbook example of \text\((x+10)(x+10)\)

, we multiply each term from the first binomial by each term from the second binomial. This is where the FOIL method comes into play, a mnemonic for multiplying First, Outer, Inner, and Last terms of each binomial. After applying FOIL to our example, we get \text\(x^2 + 10x + 10x + 100\), which simplifies to \text\(x^2 + 20x + 100\) upon combining like terms.

When the binomials are identical, as they are in the given example, this is also referred to as 'squaring a binomial', a specific case of binomial multiplication.

Simplifying Algebraic Expressions

Once you've used the FOIL method to expand binomials, the next step is simplifying the algebraic expression. Simplification can involve combining like terms, which are terms that have the same variables raised to the same powers. For our example, after applying FOIL, we get two middle terms \text\(10x + 10x\), which we combine to simplify the expression to \text\(x^2 + 20x + 100\).

The purpose of simplifying expressions is to make them easier to understand and work with, especially when solving equations or further manipulating algebraic expressions.

Squaring Binomials

Squaring binomials is a specific type of binomial multiplication where the same binomial is multiplied by itself. The term 'squaring' comes from the mathematical operation of raising a number to the power of two. When you square the binomial \text\((x+10)\), you multiply it by itself to get \text\((x+10)(x+10)\), which we've already seen.

Remember, the general formula for squaring a binomial \text\((a+b)^2\) is \text\(a^2 + 2ab + b^2\), which perfectly matches our original problem once we set \text\(a = x\) and \text\(b = 10\). Thus, squaring binomials is a handy shortcut, especially when dealing with complex or lengthy expressions.

Algebraic Equations

As you progress in algebra, you'll encounter algebraic equations, which are statements that two expressions are equal, often containing one or more variables. The goal is to solve for the unknown variable(s). After multiplying and simplifying binomials, as in the earlier steps, the result might be part of an algebraic equation you need to solve.

For example, if our simplified expression \text\(x^2 + 20x + 100 = 0\), we would then use methods like factoring, completing the square, or applying the quadratic formula to solve for \text\(x\). Algebraic equations can range from simple, one-step equations to more complex ones involving multiple steps and operations.