Problem 51 - Algebra 2

Question

Simplify. $$ (a-b)^{2} $$

Step-by-Step Solution

Answer

The simplified form is \(a^2 - 2ab + b^2\).

Step 1: Identify the Formula

To simplify the expression \((a-b)^2\), we need to recognize it as a binomial square, which can be expanded using the formula \((x-y)^2 = x^2 - 2xy + y^2\).

Step 2: Expand Using the Formula

Apply the binomial square formula to our expression by setting \(x = a\) and \(y = b\). This gives us \((a-b)^2 = a^2 - 2ab + b^2\).

Step 3: Write the Expanded Expression

The expanded form of \((a-b)^2\) is \(a^2 - 2ab + b^2\). This is the simplified version of the original expression.

Key Concepts

Binomial Expansion

The binomial expansion is a method used to simplify expressions raised to a power. In our exercise, we're dealing with the square of a binomial, specifically \( (a-b)^2 \). The binomial theorem provides a formula for expanding expressions of the form \( (x-y)^2 \), which gives us \( x^2 - 2xy + y^2 \). This means that you take the first term squared, twice the product of both terms, and the second term squared.

Understanding the pattern this theorem follows can simplify expressions and reveal their expanded forms quickly. It's particularly useful because remembering this formula for the square helps in avoiding tedious multiplications.

• First term squared: \( a^2 \)
• Twice the product of both terms: \( -2ab \)
• Second term squared: \( b^2 \)

This breakdown makes it easier to handle more complex algebraic expressions just by knowing the pattern.

Polynomial Simplification

Polynomial simplification is the process of making a polynomial expression easier to use and understand by reducing it to its simplest form. For the expression \( (a-b)^2 \), the binomial expansion has already simplified it to the polynomial \( a^2 - 2ab + b^2 \). Simplifying involves combining like terms, canceling terms that add to zero, and arranging them in standard form.

This particular expression doesn't have terms that can be further combined, so \( a^2 - 2ab + b^2 \) is already in its simplest form. Simplification helps in understanding and solving equations, minute intricacies are often hidden within more complex forms, making this skill extremely useful.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the foundation of algebra, which allows us to describe and solve problems related to numbers.

In this exercise, \( (a-b)^2 \) is an algebraic expression consisting of two variables \(a\) and \(b\), along with subtraction and exponentiation operations.

By expanding and simplifying, we demonstrate how these mathematical objects can be transformed and understood more deeply. Knowing how to manipulate algebraic expressions is a crucial tool.

• Expression without numbers: Relation of variables
• Simplified through known formulas
• Algebraic skills reduce complexity in problem-solving

These expressions form the building blocks of complex mathematical modeling and real-world problem-solving scenarios.