Problem 17 - Elementary Algebra

Question

Find the greatest common factor. \(35 x^{3}, 10 x^{4}, 5 x^{5}\)

Step-by-Step Solution

Answer

The greatest common factor is \(5x^{3}\).

Step 1: Identify the coefficients

Start by identifying the coefficients of the given terms: 35, 10, and 5.

Step 2: Find the GCF of the coefficients

Determine the greatest common factor (GCF) of the coefficients 35, 10, and 5. The GCF of these numbers is 5.

Step 3: Identify the variables and their exponents

List the variables and their exponents: \(x^{3}, x^{4}, x^{5}\).

Step 4: Find the GCF of the exponents

Identify the smallest exponent among them, which is 3.

Step 5: Combine the GCF of coefficients and exponents

Combine the GCF of the coefficients (5) and the smallest exponent (\(x^{3}\)). The greatest common factor is \(5x^{3}\).

Key Concepts

coefficients

Coefficients are the numerical parts of terms that include variables. In mathematical expressions, a number directly multiplying a variable is called a coefficient. For example, in the term \(35x^3\), \textbf{35} is the coefficient. Understanding coefficients is crucial to solving problems that involve finding the greatest common factor (GCF).

• In the problem given: 35, 10, and 5 are the coefficients.
• When we talk about the GCF of coefficients, we aim to find the largest number that divides all coefficients without leaving a remainder.

This is why it’s essential to identify coefficients correctly. Misidentifying them could lead to incorrect GCF calculations.

exponents

Exponents represent the number of times a number or variable is multiplied by itself. In the term \(x^3\), the \textbf{3} is the exponent, indicating that \(x\) is multiplied by itself three times (i.e., \(x \times x \times x\)). In the context of the problem:

• We have the exponents in \(x^3\), \(x^4\), and \(x^5\) in the terms given.
• To find the GCF of exponents, we need to find the smallest exponent because this will be the highest power of \(x\) that is common in all terms.

In our problem, the smallest exponent is 3. Hence, the common factor here is \(x^3\). This ensures that your calculations and final GCF include the correct power of the variable.

GCF calculation

Finding the greatest common factor (GCF) involves several steps. First, identify the coefficients and exponents in the expression. Then, calculate the GCF separately for the coefficients and the exponents.

• Step 1: Identify the coefficients - 35, 10, and 5.
• Step 2: Calculate the GCF of these coefficients. The GCF of 35, 10, and 5 is 5.
• Step 3: Identify the exponents - 3, 4, and 5 for the variable \(x\).
• Step 4: Calculate the GCF of the exponents, which is the smallest exponent, hence 3.
• Step 5: Combine the GCF of the coefficients (5) with the GCF of the exponents (\(x^3\)).

By combining these factors, we get the final GCF of the terms: \(5x^3\). This systematic approach ensures accurate results and helps you understand each step deeply.