Problem 20 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ 15 y^{2}, 5 y^{7},-20 y^{3} $$

Step-by-Step Solution

Answer

The GCF is \(5y^2\).

Step 1: Identify the coefficients

The coefficients of each term are 15, 5, and -20. We will find the greatest common factor (GCF) of these numbers.

Step 2: Find the GCF of the coefficients

The numbers 15, 5, and -20 have prime factorization as follows: 15 = 3 × 5, 5 = 5, and -20 = -1 × 2^2 × 5. The common factor is 5. Therefore, the GCF of the coefficients is 5.

Step 3: Identify the variables

The variable part of each term is given by the powers of \(y\): \(y^2, y^7, y^3\). The GCF will involve the smallest power of any common variable factor in all the terms.

Step 4: Determine the GCF of the variables

The smallest power of \(y\) among \(y^2, y^7, y^3\) is \(y^2\). Therefore, the GCF of the variable part is \(y^2\).

Step 5: Combine the GCF of coefficients and variables

Combine the GCF of the coefficients and the variables: 5 (from coefficients) and \(y^2\) (from variables). Thus, the overall GCF is \(5y^2\).

Key Concepts

Prime Factorization

Prime factorization is a process of breaking down a number into its prime factors. Prime factors are the prime numbers that multiply together to give the original number. For example, in the case of 15, the prime factors are 3 and 5 because 3 × 5 = 15.

Each number can be uniquely expressed as a product of prime numbers. This method is crucial when you are finding the Greatest Common Factor (GCF) among several numbers. By identifying each number's prime factors, you can find the common factors between them.

• 15 can be factorized into 3 × 5
• 5 is a prime number itself
• -20 can be factorized into -1 × 2^2 × 5

While working with negative numbers, such as -20, the negative sign can be ignored when finding the GCF because GCF is a positive number. This systematic breakdown simplifies the process of finding common factors quickly, particularly when dealing with multiple numbers.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, or division. It's like a recipe that tells us how numbers and variables interact with one another. For instance, in the expression 15**y**², you see a number (15) and a variable (**y**) raised to a power (²).

Understanding each component of an algebraic expression is essential. The **variable** represents unknown values which can change, while the **coefficient** is the constant that multiplies the variable. In 15**y**², 15 is the coefficient. When you work with GCF in algebraic expressions, you're both looking at the coefficients and the variables separately. By focusing on each part separately, it becomes easier to identify the common factors that apply to the entire expression. So, in our example with 15**y**², 5**y**⁷, and -20**y**³, you'll deal with 15, 5, -20 (coefficients), and **y**², **y**⁷, **y**³ (variables) separately to find the GCF.

Polynomial Factors

Polynomial factors are expressions that can be multiplied together to form a polynomial. In simpler terms, if you think of a polynomial as a cake, the factors are the ingredients that go into making that cake.

When solving problems like finding the GCF, it's essential to treat each part of the polynomial separately. Focus on coefficients and variables, much like you would split cake ingredients into dry and wet. For the polynomial terms 15**y**², 5**y**⁷, and -20**y**³, consider:

• The coefficients 15, 5, and -20
• The variables **y** and their corresponding powers **y**², **y**⁷, **y**³

This approach ensures that you can clearly see how each part contributes to the polynomial and makes it easier to combine like terms, such as finding a common factor. In this problem, recognizing factors involves acknowledging both the constant multipliers (numerical coefficients) and the powers of the variables, focusing on the smallest power common to all terms: **y**². By identifying the smallest shared exponent in each term, as we did with **y**², you easily determine the GCF for the polynomial terms.