Problem 21 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$2\left(10 g^{3}+5 g^{2}+4\right)-\left(2 g^{3}-14 g-20\right)$$

Step-by-Step Solution

Answer

The simplified expression is: \(18g^3 + 10g^2 + 14g + 28\).

Step 1: Distribute the 2 to the first parentheses group

Multiply each term in the first parentheses group by 2: \(2(10g^3)\), \(2(5g^2)\), and \(2(4)\).

Step 2: Distribute the negative sign to the second parentheses group

Multiply each term in the second parentheses group by -1: \(-1(2g^3)\), \(-1(-14g)\), and \(-1(-20)\).

Step 3: Execute the multiplication operations

Perform each multiplication in step1 and step2: \(20g^3\), \(10g^2\), \(8\), \(-2g^3\), \(14g\), and \(20\).

Step 4: Combine all terms

Now, combine the terms from step 3: \(20g^3 + 10g^2 + 8 - 2g^3 + 14g + 20\).

Step 5: Collect like terms

Combine all the like terms: \(20g^3 - 2g^3\), \(10g^2\), \(14g\), and \(8 + 20\).

Step 6: Simplify the expression

Simplify the expression from step 5: \(18g^3 + 10g^2 + 14g + 28\). The simplified expression is: \(18g^3 + 10g^2 + 14g + 28\).

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that allows us to multiply a single term across multiple terms inside a parenthesis. It's like handing out a gift to each person in a group. For example, in the expression \(2(10g^3 + 5g^2 + 4)\), the number 2 is distributed to each term inside the parenthesis:

• You multiply 2 by \(10g^3\) to get \(20g^3\).
• Then, multiply 2 by \(5g^2\) to get \(10g^2\).
• Finally, multiply 2 by 4 to get 8.

You must distribute to each term, not just some, ensuring the operation covers every component in the group. This process helps us break down more complex problems into manageable parts, making the algebra easier to solve.

Like Terms

Like terms refer to terms in an expression that have the same variable raised to the same power. They can be combined because they represent similar quantities. For instance, in the expression \(20g^3 - 2g^3 + 10g^2\), the terms \(20g^3\) and \(-2g^3\) are like terms because they both involve \(g^3\).

• Like terms can be added or subtracted from one another.
• In our example, \(20g^3 - 2g^3\) simplifies to \(18g^3\).

Combining like terms is vital to simplifying expressions, as it reduces them to their simplest form. Remember, only terms with the exact same variables and exponents can be combined.

Simplification

Simplification is the process of reducing an expression to its cleanest, simplest form. It involves combining like terms and applying operations to reduce the expression's complexity. In our example, after we distributed and combined the like terms, we moved to simplification.
For the expression \(20g^3 + 10g^2 + 8 - 2g^3 + 14g + 20\), simplification means:

• First, combine \(20g^3\) and \(-2g^3\) to get \(18g^3\).
• Reorder or recount the other terms like \(10g^2\) and \(14g\), which remain unchanged.
• Add constants: 8 and 20 result in 28.

By simplifying, we clarify and reveal the expression's overall gist, making it more understandable and easier to work with.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition and multiplication) that represent a specific value or relationship. They are the building blocks of algebra and provide a means to generalize mathematical ideas using symbols.
In the exercise, the expression \(2(10g^3 + 5g^2 + 4) - (2g^3 - 14g - 20)\) is an algebraic expression comprising multiple terms, both inside and outside parentheses.

• These expressions enable us to solve real-world problems by modeling relationships or changes.
• Variables like \(g\) allow expressions to be adaptable to various scenarios, not just fixed numbers.

Understanding how to manipulate algebraic expressions, whether through distributive property or combining like terms, equips students with skills to tackle various mathematical challenges.