Problem 59 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ 9 x^{3}-4 x^{3}-2 $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(5 x^{3}-2\).

Step 1: Identifying like terms

In the expression \(9 x^{3}-4 x^{3}-2\), the like terms are \(9 x^{3}\) and \(-4 x^{3}\). These two terms can be combined because they both have the same variable part, \(x^{3}\).

Step 2: Combining like terms

Combine \(9 x^{3}\) and \(-4 x^{3}\) to get \(5 x^{3}\).

Step 3: Writing the simplified expression

The simplified expression after combining like terms is \(5 x^{3}-2\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental process that helps simplify expressions. Like terms are terms whose variables, and their exponents, are the same. When you see an expression like \(9 x^{3} - 4 x^{3} - 2\), you can combine the terms with the same variable raised to the same power. In this expression, \(9 x^{3}\) and \(4 x^{3}\) are like terms because they both have the variable \(x\) raised to the third power. They can be combined by adding their coefficients (the numbers in front of the variables) together: \(9 - 4\) yields \(5\), and because the variable parts are the same, we get \((9-4)x^{3} = 5x^{3}\).

The constant term \(2\), which has no variable attached, is left unchanged because it does not have a like term to combine with. So, the simplified expression is \(5 x^{3} - 2\). Remember, only terms with the exact same variable (including the variable's exponent) can be combined. It's like bundling similar items together, simplifying the expression.

Simplifying Expressions Step by Step

To simplify an algebraic expression step by step, it's helpful to follow a systematic approach. Let's walk through the process using the expression \(9 x^{3}-4 x^{3}-2\):

• Step 1: Identify the like terms, which are terms that contain the same variable raised to the same power. Here, we focus on \(9 x^{3}\) and \(4 x^{3}\).
• Step 2: Combine the like terms by adding or subtracting their coefficients. We do \(9-4\) to get \(5\), so combined they are \(5x^{3}\).
• Step 3: Rewrite the expression with the combined like terms and any remaining terms. We conclude with the simplified expression \(5 x^{3} - 2\).

By taking these steps one by one, you ensure that you don't miss out on combining any terms, and you incrementally simplify the expression until it's fully reduced. This systematic approach is especially useful for more complex expressions with multiple like terms and variables.

Algebraic Terms

An algebraic term is a combination of numbers, variables, and sometimes exponents that are multiplied together. Terms can be simple, like \(5\) or \(x\), or more complex, such as \(9x^{3}\) or \(2xy^2\). The key parts of an algebraic term are the coefficient and the variable part. For the term \(9x^{3}\), \(9\) is the coefficient, which is a numeric factor, while \(x^{3}\) is the variable part, indicating that \(x\) is the variable raised to the third power.

In isolation, terms aren't very telling, but when they come together in an expression, they can be powerful in representing numbers and relationships. Deciphering these terms includes understanding how they interact and combine, which is crucial for simplifying expressions and solving equations. Recognize that constants are terms too, but they stand alone without variables, and they play a role as the canvas of the algebraic world - they give us fixed points of reference.