Problem 15 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ 12 y^{4}, 20 y^{3} $$

Step-by-Step Solution

Answer

The GCF is \(4y^3\).

Step 1: Factor Each Term

Start by factoring each term into prime factors and separating out any variable parts.For \(12y^4\):\(12 = 2^2 \times 3\) and \(y^4 = y \times y \times y \times y\)So, \(12y^4 = 2^2 \times 3 \times y^4\).For \(20y^3\):\(20 = 2^2 \times 5\) and \(y^3 = y \times y \times y\)So, \(20y^3 = 2^2 \times 5 \times y^3\).

Step 2: Identify Common Factors

Compare the factored forms from Step 1 and identify the common factors.From \(12y^4 = 2^2 \times 3 \times y^4\) and \(20y^3 = 2^2 \times 5 \times y^3\), the common factors are \(2^2\) and \(y^3\).

Step 3: Calculate the GCF

Multiply the common factors to get the GCF.The GCF is \(2^2 \times y^3\). Calculating that gives \(4y^3\).

Key Concepts

Prime Factorization

Prime factorization is a process used to break down numbers into their basic building blocks. These building blocks are the prime numbers. A prime number is a number that can only be divided by one and itself without leaving a remainder.
For example, when we perform prime factorization on the number 12, we break it down into smaller factors:

• Divide the number by the smallest possible prime: 12 divided by 2 gives 6, and 6 can be further divided by 2 to give 3.
• The final result is that 12 is composed of the prime numbers 2, 2, and 3 (written as \(2^2 \times 3\)).

This means that the prime factorization of 12 is \(2^2 \times 3\).
When factoring polynomial expressions, we apply the same process to any numerical coefficients, while treating variables separately.
In our example, for having polynomial terms such as \(12y^4\) and \(20y^3\), each number is broken into its prime factors first, and then variables are treated based on their powers, or exponents. This method of focusing on prime factors helps us identify common elements in problems like finding the greatest common factor.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables, each multiplied by coefficients. These expressions are commonplace in algebra and beyond.
Here's what you need to know about polynomial expressions:

• They can include constants, variables, and exponents where variables are raised to whole-number powers.
• The simplest polynomial expression is a monomial, which consists of just one term, such as \(12y^4\).
• Polynomials can also consist of multiple terms, and these terms are joined by addition or subtraction.

In our exercise, each term of the polynomial is evaluated separately. The terms like \(12y^4\) and \(20y^3\) are essentially polynomial expressions that we simplify individually to find common factors.
Polynomials form the basis of algebraic manipulation and are key to solving equations and understanding deeper algebraic concepts.

Algebraic Terms

Algebraic terms form the components of polynomial expressions. Each term is made up of several factors:

• A numerical coefficient, which is the numerical part of the term.
• Variables, which may have exponents attached to them.

For instance, in the algebraic term \(12y^4\), 12 is the coefficient, while \(y^4\) is the variable part.
This indicates that the variable y is multiplied by itself four times.
Algebraic terms are combined using addition or subtraction to construct polynomial expressions.
In our example, recognizing the structure of each term allows us to apply concept of the greatest common factor to simplify expressions.
When solving problems, breaking each term into its individual components (as seen with prime factorization) provides clarity and simplification. This method of analysis aids in performing calculations that would otherwise seem complex, such as finding the greatest common factor, or GCF.