Problem 6 - Algebra A Combined Function

Question

$$ \left(3 x^{2}+7\right)+\left(3 x^{2}+9\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^2 + 16\).

Step 1: Understand the Expression

The given expression is \((3x^2 + 7) + (3x^2 + 9)\). It involves the addition of two polynomial expressions.

Step 2: Combine Like Terms

Add the like terms from both expressions. The like terms here are the constant terms and the \(x^2\) terms. So, we have:\[3x^2 + 3x^2 = 6x^2\]\[7 + 9 = 16\]

Step 3: Write the Final Expression

Combine the results from the previous step to form the simplified expression: \[6x^2 + 16\]

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental skill when working with expressions. Like terms are terms that have the same variable raised to the same power. This means they can be combined through addition or subtraction to simplify the expression.

For instance, in the expression \((3x^2 + 7) + (3x^2 + 9)\), the terms containing \(x^2\) are considered like terms. These are \(3x^2\) from each part of the expression. By combining these, you can add them together:

• \(3x^2 + 3x^2 = 6x^2\)

Like terms also apply to constant numbers. So, in our example, the numbers 7 and 9 are combined:

• \(7 + 9 = 16\)

This step is crucial for simplifying algebraic expressions and finding a clearer form of the problem you are working on.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form. This makes it easier to work with and understand polynomials or any algebraic expressions.

To simplify an expression, you start by performing any operations that are indicated, such as addition or subtraction, and apply the rules of arithmetic where possible. Remember, the goal is to make the expression as simple as possible without changing its value.

In our example of simplifying \((3x^2 + 7) + (3x^2 + 9)\), after combining the like terms, we ended up with:

• \(6x^2 + 16\)

This expression is simpler because all similar terms are combined, and there are no more operations to perform. This streamlined form is often more useful for further calculations or solving equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations (like addition and multiplication). Understanding how to manipulate these expressions is key to mastering algebra.

Each part of an algebraic expression is called a term. For example, in \(3x^2 + 7\), there are two terms: \(3x^2\) and \(7\). Algebraic expressions can be simplified, combined, and even solved for specific values, depending on the problem.

Working with algebraic expressions often involves processes such as:

• Identifying like terms to combine them
• Rearranging terms to simplify
• Applying arithmetic operations to simplify

Mastering algebraic expressions is fundamental to progressing in more advanced mathematics, making these basic skills incredibly valuable for students. By consistently identifying and manipulating parts of these expressions, you gain a deeper understanding and build a critical math foundation.