Question

Which of the following expressions is equivalent to \(2 a(a\) \(\left.-3 b^2\right)+a^2 ?\) A. \(2 a^2-6 a b^2\) B. \(3 a^2-3 b^2\) C. \(2 a\left(a-3 b^2\right)\) D. \(3 a\left(a-2 b^2\right)\)

Short answer

The short answer is A: \(3a^2 - 6ab^2\).

Step-by-step solution

Distribute the constant in the expression

First, distribute the 2 inside the parenthesis: \[2a(a - 3b^2) + a^2 = 2a^2 - 6ab^2 + a^2\]

Combine like terms

Now, combine the like terms (both of which are \(a^2\)): \[2a^2 - 6ab^2 + a^2 = 3a^2 - 6ab^2\]

Compare the simplified expression with given options

Compare the simplified expression \(3a^2 - 6ab^2\) with the given options: A. \(2 a^2 - 6 a b^2\) B. \(3 a^2 - 3 b^2\) C. \(2 a(a - 3 b^2)\) D. \(3 a(a - 2 b^2)\) Our simplified expression matches option A, so the correct answer is A: \(2 a^2 - 6 a b^2\).

Key concepts

Distributive Property

The distributive property is a key concept in algebra that allows you to simplify expressions by "distributing" a factor across terms inside parentheses. The distributive property follows this form:

• If you have an expression like \(a(b + c)\), you can rewrite it as \(ab + ac\).

In our problem, we start with \(2a(a - 3b^2) + a^2\). Here, you need to "distribute" the \(2a\) across \(a\) and \(-3b^2\). This means:

• Step 1: Multiply \(2a\) by \(a\) to get \(2a^2\).
• Step 2: Multiply \(2a\) by \(-3b^2\) to get \(-6ab^2\).

So, the expression inside the parentheses becomes \(2a^2 - 6ab^2\). Always ensure to multiply each term inside the parentheses by the term outside to correctly apply the distributive property.

Combining Like Terms

Combining like terms is a method in algebra used to simplify expressions and make calculations easier. Like terms are terms that have identical variable parts. To effectively combine them, you simply add or subtract their coefficients.

• In our simplified expression, namely \(2a^2 - 6ab^2 + a^2\), notice that both \(2a^2\) and \(a^2\) have the variable part \(a^2\). That's what makes them 'like terms.'

By adding their coefficients, we get:

• 2 (from \(2a^2\)) plus 1 (from \(a^2\)) gives us \(3a^2\).

This combination effectively produces a more manageable expression: \(3a^2 - 6ab^2\). Combining like terms is crucial for simplifying expressions, helping ones work with fewer terms.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic and concise form. The goal is to have an expression that is easier to understand and work with without changing its value.

• Start with the expanded or distributed form, like \(2a^2 - 6ab^2 + a^2\), and use algebraic techniques, like combining like terms, to simplify.

After applying the distributive property and combining like terms, your expression should be in its simplest form: \(3a^2 - 6ab^2\). Always ensure your final expression has no like terms left to combine and represents the same quantity as the original expression.Comparing the final simplified expression to provided options helps confirm that it is truly in its simplest form. By consistently practicing how to simplify, you master making intricate algebraic problems more approachable and solvable.