Problem 56 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((r-2 s)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(r^2 - 4rs + 4s^2\).

Step 1: Identify the Formula

The expression \((r - 2s)^2\) is a square of a binomial. This fits the special product formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = r\) and \(b = 2s\).

Step 2: Apply the Formula

Substitute \(a\) and \(b\) into the formula: \((r - 2s)^2 = r^2 - 2(r)(2s) + (2s)^2\).

Step 3: Calculate Each Term

Compute each part of the formula: - \(r^2\) is simply \(r^2\).- \(-2(r)(2s) = -4rs\).- \((2s)^2 = 4s^2\).

Step 4: Write the Expression

Combine the calculated terms: The expression simplifies to \(r^2 - 4rs + 4s^2\).

Key Concepts

Binomial Square

When you encounter a binomial, you are dealing with an expression that contains two terms. In this case, our binomial is \((r-2s)\). Squaring a binomial means multiplying the binomial by itself. The expression becomes \((r-2s) \times (r-2s)\). However, there's a quicker way to solve this using the binomial square formula:

• For a binomial squared, the formula is \((a-b)^2 = a^2 - 2ab + b^2\), where each term of the binomial is squared, and twice their product is also included.
• In our exercise with \((r-2s)^2\), you identify \(a=r\) and \(b=2s\).
• Applying the formula can significantly simplify your work and avoid potential calculation errors in basic algebra.

Understanding and using the binomial square formula properly can help make these calculations faster and more accurate.

Algebraic Expressions

Algebraic expressions, like \((r-2s)^2\), allow us to describe relationships between different quantities using variables and numbers.
These expressions can include operators such as addition, subtraction, multiplication, and division, and they frequently incorporate parentheses to dictate the order of operations.

Each component in an algebraic expression plays a unique role:

• Variables represent unknown quantities and are typically denoted by letters like \(r\) and \(s\).
• Coefficients, like the \(2\) in \(2s\), multiply the variables and represent known quantities.
• Constants are fixed values that appear without variables, but there are none in this particular expression.

Being comfortable with algebraic expressions is fundamental, as they are essential in formulating equations and solving for unknowns in mathematics.

Simplifying Expressions

Simplifying expressions involves reducing them to a simpler form or a particular type of finished form.
Simplifying \((r-2s)^2\) requires following a sequence of operations by applying the special product formula.

Key steps include:

• Apply the formula: Start by using \((a - b)^2 = a^2 - 2ab + b^2\) which gives you a roadmap for what the simplified expression should look like.
• Compute each term: First, find \(a^2\), which is \(r^2\). Then \(-2ab\ = -4rs\), and finally \(b^2 = 4s^2\).
• Combine terms: Once computed, put \(r^2\), \(-4rs\), and \(4s^2\) together to simplify the expression to \(r^2 - 4rs + 4s^2\).

Simplifying expressions makes them easier to read and understand, allowing mathematicians to communicate ideas effectively and solve equations with ease.