Problem 54 - Elementary Algebra

Question

Simplify \(3 a^{2}-2 a+4 a(a+2)\).

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(3a^2 - 2a + 4a(a+2)\). Answer: \(\boxed{7a^2 + 6a}\)

Step 1: Distribute multiplication in the third term

We start by expanding the third term in the expression: \(4a(a+2)\). To do this, distribute the \(4a\) by multiplying it to both terms inside the parentheses: $$4a \cdot a + 4a \cdot 2$$ This results in: $$4a^2 + 8a$$ So now our original expression looks like this: $$3a^2 - 2a + (4a^2 + 8a)$$

Step 2: Combine like terms

Now, we need to group like terms and add their corresponding coefficients: $$3a^2 + 4a^2 - 2a + 8a$$ We now have: $$7a^2 + 6a$$ Hence, the simplified expression is \(\boxed{7a^2 + 6a}\).

Key Concepts

Distribute Multiplication

In algebra, distribute multiplication is a crucial technique used to simplify expressions. It involves spreading or "distributing" a factor over terms within a set of parentheses. This step effectively removes the parentheses by multiplying each term inside by the factor outside.

For instance, in the expression \(4a(a+2)\), we distribute the \(4a\) over both terms \(a\) and 2. This means we multiply \(4a \cdot a\) and \(4a \cdot 2\).

• \(4a \cdot a = 4a^2\)
• \(4a \cdot 2 = 8a\)

When distributing, it's essential to perform the operation on each term independently to ensure that none of the terms are missed. This allows us to transform the expression from one with parentheses into one that's a simple sum, ultimately giving us \(4a^2 + 8a\). By mastering this method, we can handle more complex polynomials confidently.

Combine Like Terms

Once distribution has been applied, the next step in simplifying algebraic expressions is to combine like terms. Like terms are those that have the same variable raised to the same power. In our example, \(3a^2 + 4a^2 - 2a + 8a\), \(3a^2\) and \(4a^2\) are like terms, as are \(-2a\) and \(8a\).

Combining like terms means adding or subtracting their coefficients:

• \(3a^2 + 4a^2 = 7a^2\)
• \(-2a + 8a = 6a\)

The importance of combining like terms is to reduce the expression to its simplest form. This involves collecting all coefficients of similar terms and merging them into one, streamlining the original expression to \(7a^2 + 6a\).

This process not only simplifies the expression but also makes it easier to interpret and solve, especially when dealing with equations. Properly combining like terms can turn a complex jumble of numbers and variables into a clear and concise polynomial.

Algebraic Expressions

Understanding algebraic expressions is fundamental in algebra. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions can represent real-world quantities and relationships in mathematics. For example, the expression \(3a^2-2a+4a(a+2)\) is composed of constants, variables, and operations.

Algebraic expressions can range from simple to highly complex. Simplifying them involves several steps including distributing multiplication and combining like terms as illustrated in the previous sections.

• Constant: A number on its own, such as 3 or 4.
• Variable: A symbol, like \(a\), that represents a number we don't know yet.
• Coefficient: A number that multiplies a variable, like 3 in \(3a^2\).
• Term: A single mathematical expression, like \(3a^2\) or \(-2a\).

By understanding and applying techniques to simplify these expressions, we can often find solutions to complex problems more efficiently. Grasping the concept of algebraic expressions really means mastering a powerful tool for both academic challenges and real-life scenarios.