Problem 7 - Elementary Algebra

Question

List, if any appear, the common factors in the following expressions. $$ x^{2}+5 x^{2}-9 x^{2} $$

Step-by-Step Solution

Answer

Answer: The common factors are -1, 3, -3, x, and \(x^2\).

Step 1: Combine like terms

To combine like terms, add or subtract the coefficients of the terms with the same power of x: $$ x^2 + 5x^2 - 9x^2 = (1 + 5 - 9)x^2 = -3x^2 $$

Step 2: Identify the common factors

Now that the expression is simplified to \(-3x^2\), we can see that there are common factors. The factors of this expression are -1, 3, -3, x, and \(x^2\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a key step in simplifying expressions. This involves grouping together terms that have the same variable raised to the same power. For example, in the expression \(x^2 + 5x^2 - 9x^2\), all terms have the variable \(x\) raised to the power of 2. To combine these, you add or subtract their coefficients (the numerical parts).

• The first term is \(1x^2\), the second is \(5x^2\), and the third is \(-9x^2\).
• Thus, combining them involves solving the equation \(1 + 5 - 9\), which results in \(-3\).
• Hence, the simplified expression is \(-3x^2\).

This process is crucial as it reduces complex expressions into simpler forms. It helps you understand and solve equations more easily.

Common Factors

Finding common factors within an algebraic expression is an essential skill. This process involves identifying the quantities that can divide every term in the expression without a remainder. For \(-3x^2\), the factors include:

• -1,
• 3,
• -3,
• x,
• and \(x^2\).

Each of these factors can evenly divide the expression. Recognizing common factors is particularly useful for factoring expressions, solving equations, and simplifying terms. It allows you to break down complex equations which can then be more easily worked with or interpreted in further mathematical tasks.

Polynomial Simplification

Simplifying polynomials involves combining like terms and identifying common factors, which are the crucial steps in reducing a polynomial into its most concise form. When you simplify a polynomial, it becomes easier to work with, especially during additional operations such as adding or multiplying other polynomials.

• For the expression \(-3x^2\), simplification has already been achieved by combining like terms.
• This results in a singular term that captures the essence of the original polynomial.
• Additionally, recognizing the common factors in this simplified expression further aids in potential future factorization.

The simplified form provides clarity. It reveals the inherent relationships or operations within the polynomial itself and facilitates further mathematical operations.