Question

Expand \((a+b+c)^{2}\)

Short answer

The expanded expression is \(a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca\).

Step-by-step solution

Identify the Expression

The expression we need to expand is \[(a+b+c)^{2}\]This is a binomial expression, and it needs to be expanded using the distributive property or a known algebraic identity.

Use the Expansion Formula

Use the formula for expanding a squared trinomial: \[(a+b+c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca\]This formula will guide the expansion process.

Apply the Formula to Each Term

Apply the formula step-by-step:- First, square each term: - \(a^{2}\) - \(b^{2}\) - \(c^{2}\)- Next, find each pairwise product and multiply by 2: - \(2ab\) - \(2bc\) - \(2ca\)

Write the Expanded Expression

Combine all terms obtained in the previous step. The expanded form of the expression is:\[a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca\]

Key concepts

Distributive Property

In algebra, the distributive property is a key principle that helps us manage expressions involving multiplication and addition. This property allows us to break down complex expressions into more manageable pieces, especially when expanding or simplifying algebraic expressions. It states that for any numbers or expressions, you distribute the multiplication over addition:

• \(a(b + c) = ab + ac\)

This means you can multiply each part of the expression inside the parentheses by the term outside. This property is particularly helpful when dealing with expressions like \((a + b + c)^2\), where each term in the binomial needs to be squared and each pair of terms multiplied together. Applying the distributive property efficiently can simplify complex polynomial expansions and is an essential skill in algebra.

Algebraic Identity

Algebraic identities are special equations that hold true for all values of the variables included. They provide formulas or rules, helping to simplify algebraic expressions. One of the most common identities is the formula for the square of a trinomial, such as \((a + b + c)^2\). This identity can be written as:

• \[(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\]

This is more than just a formula—it's a powerful tool to expand expressions systematically. By applying this identity, you eliminate guesswork and can quickly expand any squared trinomial. Understanding and using algebraic identities effectively helps you solve problems faster and more accurately, as shown in this exercise.

Squared Trinomial

The concept of a squared trinomial involves taking a trinomial expression—one with three terms—and squaring it. This operation creates a polynomial with specific characteristics. The expression \((a + b + c)\) represents a trinomial, and squaring it involves expanding it using a known formula:

• \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)

A squared trinomial results in nine distinct operations: three squaring operations for each term, and six multiplication operations for each pair of terms, each summed up with a factor of two due to their symmetry. Understanding such structures and the resulting polynomial can help you solve and expand trinomials efficiently. Using these strategies makes complex algebraic expressions simpler to manage and lays the foundation for more advanced problem-solving in algebra.