Problem 22 - Elementary Algebra

Question

Find the following products and simplify. $$ (r-7)(r-7) $$

Step-by-Step Solution

Answer

Answer: The simplified product of (r-7)(r-7) is r^2 - 14r + 49.

Step 1: Distribute the first term

We begin by multiplying the first term, \(r\), inside the first binomial with both terms in the second binomial: \((r)(r-7) + (-7)(r-7)\). This will give us: $$ r^2 - 7r - 7(r-7) $$

Step 2: Distribute the second term

Now, let's distribute the second term, \(-7\), inside the first binomial with both terms in the second binomial: $$ r^2 - 7r - 7r + 49 $$

Step 3: Combine like terms

Lastly, we will combine the like terms which are the two middle terms(-7r and -7r): $$ r^2 - 14r + 49 $$ So, the simplified product of \((r-7)(r-7)\) is: $$ (r-7)(r-7) = r^2 - 14r + 49 $$

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing an expression to its simplest form by performing operations and combining like terms. When you encounter expressions like \( (r-7)(r-7) \), the goal is to use algebraic rules to make them as straightforward as possible. Simplifying makes expressions easier to understand and work with, especially when you're solving equations or further manipulating them. Let's explore some key steps involved:

• Perform all possible multiplications and apply the distributive property. This helps break down complicated parts of the expression.
• Combine like terms, which are terms that have the same variables and powers, such as \(-7r\) and \(-7r\) in our example.
• Reduce the expression to its simplest possible form. In our case, \( r^2 - 14r + 49 \) is simpler than the original multiplied form.

Learning to simplify expressions is crucial not just for individual problems but as a building block for more complex algebraic operations.

Binomial Expansion

The binomial expansion involves multiplying out expressions where two terms are raised to a power, expressed in the form \((a+b)^n\). It's particularly useful when dealing with expressions like \((r-7)(r-7)\). The expansion follows the rules of binomial theorem, though in simpler cases, like \((r-7)^2\), it's more about applying the distributive property as seen in the example:

• First, recognize the expression \((r-7)(r-7)\) as \((r-7)^2\).
• Apply the distributive steps: multiply each term in the first binomial by each term in the second.
• Resolve all multiplications, resulting in an expression like \(r^2 - 14r + 49\) after combining terms.

Mastering binomial expansion allows students to tackle polynomials with confidence and is a fundamental concept when progressing in mathematics.

Distributive Property

The distributive property is a basic but crucial principle in algebra that helps in multiplying expressions. It states that given three terms, \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). In the context of our example, \((r-7)(r-7)\), this property allows you to break down the multiplication into simpler parts:

• Begin by multiplying the first term, \(r\), in the first binomial by each term in the second binomial, resulting in \(r^2 - 7r\).
• Then, take the second term, \(-7\), and multiply by each term in the second binomial, giving \(-7r + 49\).
• Finally, distribute and combine these results to form a simplified expression like \(r^2 - 14r + 49\).

Understanding and using the distributive property effectively is foundational for simplifying expressions and performing polynomial multiplication.