Problem 5 - Elementary Algebra

Question

Simplify each of the following expressions. $$ 10 x^{3}-4 x^{3}+3 x^{2}-12 x^{3}+5 x^{2}+2 x+x^{3}+8 x $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: $$ 10 x^{3}-4 x^{3}+3 x^{2}-12 x^{3}+5 x^{2}+2 x+x^{3}+8 x $$ Answer: The simplified expression is: $$ -15 x^{3} + 8 x^{2} + 10 x $$

Step 1: Group like terms

According to the given expression, we have the following terms: $$ 10 x^{3}-4 x^{3}+3 x^{2}-12 x^{3}+5 x^{2}+2 x+x^{3}+8 x $$ We need to group the like terms together. We have the terms with \(x^{3}\), \(x^{2}\), and \(x\), so we rewrite the expression as follows: $$ (10 x^{3}-4 x^{3}-12 x^{3}+x^{3})+(3 x^{2}+5 x^{2})+(2 x+8 x) $$

Step 2: Add and subtract the like terms

Now, we will add and subtract the like terms to simplify the expression. For the terms with \(x^{3}\), we have $$ 10 x^{3}-4 x^{3}-12 x^{3}+x^{3} = (-4-12+1) x^{3} = -15 x^{3} $$ For the terms with \(x^{2}\), we have $$ 3 x^{2}+5 x^{2} = (3+5) x^{2} = 8 x^{2} $$ For the terms with \(x\), we have $$ 2 x+8 x = (2+8) x = 10 x $$

Step 3: Write the simplified expression

Now that we have added and subtracted the like terms, we can write the simplified expression as follows: $$ -15 x^{3} + 8 x^{2} + 10 x $$ So, the simplified expression of the given algebraic expression is: $$ -15 x^{3} + 8 x^{2} + 10 x $$

Key Concepts

Like Terms

Recognizing and working with 'like terms' is an important step in simplifying algebraic expressions. Like terms are terms in an expression that have the same variable raised to the same power. For instance, in the expression \(10x^3 - 4x^3 - 12x^3 + x^3\), all the terms contain the variable \(x^3\). This means they are like terms. By grouping like terms, you allow for easy addition or subtraction, simplifying the overall expression.

• Terms containing \(x^3\) will be grouped and simplified together.
• Similarly, for terms with \(x^2\), which are considered like terms, the process is applied.
• Also, \(x\) terms are grouped to be simplified together.

Grouping and simplifying like terms make handling expressions much easier, creating a clear path towards the solution. Understanding the nature of like terms will greatly help in ensuring that calculations are accurate and efficient.

Simplifying Expressions

Simplifying expressions is a process to make algebraic expressions less complicated. This involves combining like terms to reduce the number of terms in a polynomial. It requires careful attention to the coefficients (the numbers in front of the variables) as well as the variable parts.

The main steps to simplify expressions include:

• First, identify and group like terms within the expression.
• Next, add or subtract corresponding coefficients of these like terms.
• Finally, write the simplified expression by combining the results.

In our example, after grouping like terms, the terms with \(x^3\) simplify to \(-15x^3\), \(x^2\) terms to \(8x^2\), and \(x\) terms to \(10x\). The simplified expression thus reads as \(-15x^3 + 8x^2 + 10x\). Simplifying expressions allows for a clearer understanding and more manageable manipulation for further mathematical processes or solving.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators. Algebraic expressions can range from simple ones, like \(2x + 3\), to more complex forms, such as \(10x^3 - 4x^3 + 3x^2 - 12x^3 + 5x^2 + 2x + x^3 + 8x\).

Key elements of algebraic expressions include:

• Variables: symbols that represent numbers (e.g., \(x\))
• Coefficients: numeric factors of terms (e.g., 10 in \(10x^3\))
• Operators: mathematical signs (+, -, etc.)

Algebraic expressions serve different purposes in mathematics and are used to express relationships, model real-world scenarios, and solve various equations.

The efficient manipulation of these expressions through processes such as simplifying helps both in solving equations and understand the underlying relationships. The example expression demonstrates how understanding the structure and components of an algebraic expression can aid in its simplification, leading to a clearer and more concise result like \(-15x^3 + 8x^2 + 10x\).