Problem 2 - Elementary Algebra

Question

List, if there are any, the common factors of \(20 x^{3} y^{2}+15 x^{3} y^{2} z^{2}+10 x^{3} z^{2}\)

Step-by-Step Solution

Answer

Answer: The common factor of the given expression is \(5x^3\).

Step 1: Find the common numerical coefficients

First, we'll find the common numerical coefficients in each term. The coefficients are 20, 15, and 10. The greatest common divisor of 20, 15, and 10 is 5, so 5 is the common numerical factor.

Step 2: Find the common variables with their minimal powers

Now, let's find the common variables in each term: 1. \(x^3y^2\) 2. \(x^3y^2z^2\) 3. \(x^3z^2\) As we can see, \(x^3\) is present in all three terms. Also, note that \(y^2\) is present in the first two terms but not in the third term, so it's not a common factor. Similarly, the variable \(z^2\) is present in the second and third terms, but not in the first term, so it's not a common factor as well. Therefore, \(x^3\) is the only common variable factor.

Step 3: Combine the common numerical and variable factors

Finally, let's combine the common numerical and variable factors found in steps 1 and 2. The common numerical factor is 5, and the common variable factor is \(x^3\). Multiplying these two factors, we get the final common factor: 5\(x^3\) So, the common factor of the given expression is \(5x^3\).

Key Concepts

Common Factors

When working with polynomials, identifying common factors can simplify expressions and help solve equations more effectively. Common factors are elements that appear in each term of a polynomial. For instance, if you have a polynomial expression like \(20x^3y^2 + 15x^3y^2z^2 + 10x^3z^2\), you should first look for common numerical coefficients and variables.

Here’s how you do it:

• Identify each term's coefficient.
• Examine the variables and their powers.

In the expression above, the coefficients are 20, 15, and 10. By determining what number can divide all three coefficients evenly, you find the greatest numerical common factor, which is 5 in this case. Next, focus on the variables with the minimum powers present in all terms. For example, \(x^3\) appears in every term, making it a common factor.

By combining both the numerical common factor (5) and the variable common factor (\(x^3\)), you have 5\(x^3\). Recognizing and using common factors enhances understanding and streamlines polynomial calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept in mathematics, especially when dealing with polynomials. It refers to the largest number or expression that divides two or more numbers or expressions without leaving a remainder.

To identify the GCD among the coefficients in a polynomial, such as those from the expression \(20x^3y^2 + 15x^3y^2z^2 + 10x^3z^2\), you follow these steps:

• List the factors of each coefficient.
• Identify the largest factor common to these coefficients.

For example, the factors for 20, 15, and 10 are:

• 20: 1, 2, 4, 5, 10, 20
• 15: 1, 3, 5, 15
• 10: 1, 2, 5, 10

The common factor here is 5, which becomes the GCD. In polynomial expressions, finding the GCD of numerical coefficients and determining a common variable factor can assist in factoring the polynomial effectively. It’s an invaluable skill in algebra, aiding in problem-solving and simplification of expressions.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables raised to whole number powers and non-negative integer coefficients. They can have multiple terms, each made up of a product of a constant coefficient and variable(s) raised to powers, like the expression \(20x^3y^2 + 15x^3y^2z^2 + 10x^3z^2\).

Understanding polynomial expressions involves recognizing their structure and the relationships between terms. Key attributes include:

• Coefficient: The numerical part of a term (e.g., 20, 15, 10).
• Degree: The highest power of the variable present in any term.
• Variable: A symbol representing a number, typically denoted as \(x, y, z\).

Factoring polynomials means breaking them into simpler components, which often involves finding common factors. This not only aids in solving polynomial equations but also simplifies computational work. A solid grasp of polynomial expressions is fundamental to exploring more advanced algebraic concepts and mathematical problem-solving. By frequently practicing factoring and identifying polynomial components, students can build a strong foundation for further mathematical study.