Problem 40 - Elementary Algebra

Question

For the following problems, list, if any should appear, the common factors in the expressions. $$ 6 x^{2} y^{3}+18 x^{2} $$

Step-by-Step Solution

Answer

Answer: The common factors between the given expressions are $$2$$, $$3$$, and $$x^2$$.

Step 1: Identify the expressions

We are given the following expression: $$ 6x^2y^3 + 18x^2 $$ We can see there are two terms in the expression: 1. $$6x^2y^3$$ 2. $$18x^2$$

Step 2: Break down each term into its simplest form

Now, let's break down each term into its simplest form: 1. $$6x^2y^3 = 2\cdot3\cdot x^2\cdot y^3$$ 2. $$18x^2 = 2\cdot3\cdot3\cdot x^2$$

Step 3: Identify the common factors

The common factors between the two terms are: 1. $$2$$ 2. $$3$$ 3. $$x^2$$

Step 4: List the common factors

Now that we have identified the common factors between the two terms, let's list them: 1. $$2$$ 2. $$3$$ 3. $$x^2$$

Key Concepts

Factoring Algebraic Expressions

When we factor algebraic expressions, we are essentially breaking down a complex expression into simpler parts that, when multiplied together, give us the original expression. This is analogous to breaking down a number into its prime factors. For example, the number 18 can be factored into the prime numbers 2 and 3 (since 18 = 2 x 3 x 3). Similarly, an algebraic expression like
\(6x^2y^3 + 18x^2\) can be factored into smaller parts such as coefficients, variables, and their exponents.
Factoring is a helpful process used for simplifying expressions, solving equations, and finding zeroes of polynomials. It is especially useful in simplifying fractions and canceling terms. The first step is identifying terms within the expression which can include constants, variables, and exponents. The next step is to find the common factors that can be divided out from each term, which simplifies the expression and may reveal underlying relationships or patterns.

Greatest Common Factor

The Greatest Common Factor (GCF), also known as the greatest common divisor, is the largest factor shared by two or more numbers or terms. For instance, in the terms
\(6x^2y^3\) and \(18x^2\), we can find the GCF by breaking each term down into its prime factors and variable parts. The number 6 is made up of the prime factors 2 and 3, and for 18, the prime factors are 2, 3, and 3, which we denote as \(2 \times 3^2\).
Both terms contain the variable \(x^2\), but only the first term has the variable \(y^3\). Here, the GCF is the product of the highest power of common factors that appear in each term, including both number and variable factors. In our example, the GCF is
\(2 \times 3 \times x^2\) as these are the factors that appear in both the terms. Identifying the GCF is key when factoring polynomials, simplifying fractions, or finding the simplest form of an expression.

Polynomial Terms

Polynomials consist of terms that are algebraic expressions made up of constants, variables, and exponents. A term is a single element or a product of several elements (number, variable, or the product of numbers and variables) that are combined using multiplication. In the polynomial
\(6x^2y^3 + 18x^2\), there are two terms:
1. \(6x^2y^3\) - This term has the constant 6, the variable \(x\) raised to the second power, and the variable \(y\) raised to the third power.
2. \(18x^2\) - The second term consists of the constant 18 and the variable \(x\) raised to the second power.
The degree of a term is the sum of the exponents of the variables within it. For instance, the term \(6x^2y^3\) has a degree of 5 (since 2 + 3 = 5). Understanding the structure of polynomial terms is crucial for performing operations such as addition, subtraction, and particularly when factoring, as one usually looks for the common factors among the terms.