Problem 16 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 6 y^{2}+30 y $$

Step-by-Step Solution

Answer

The factored form is \(6y(y+5)\). The polynomial \(y+5\) is prime.

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the polynomial. The terms are \(6y^2\) and \(30y\). The coefficients are 6 and 30, and their GCF is 6. The variable part for the lowest power of \(y\) is \(y\). Therefore, the GCF is \(6y\).

Step 2: Factor Out the GCF

Factor the GCF out of each term in the polynomial. This involves dividing each term by the GCF and writing the polynomial as the product of the GCF and the resulting polynomial:\[6y^2 + 30y = 6y(y + 5).\]

Step 3: Check the Factoring

Verify the factoring by distributing \(6y\) back into the polynomial. This ensures the original polynomial is obtained:\[6y(y + 5) = 6y^2 + 30y.\]

Step 4: Identify Prime Polynomials

Examine the factored polynomial to identify prime polynomials. The polynomial \(y + 5\) cannot be factored further and is thus a prime polynomial.

Key Concepts

Greatest Common Factor (GCF)

Factoring polynomials often involves identifying the Greatest Common Factor (GCF). The GCF of algebraic expressions is the largest factor that divides each term. Here's how you can do it:

1. **Identify Numerical Coefficients**: Look at the numbers in front of the variables. For instance, in the terms 6 and 30, the GCF is 6.
2. **Identify Variable Factors with Common Powers**: Look for the lowest powers of common variables. For example, in the terms with variables, the lowest power of y in both terms is y.
Combining both steps, the GCF of the polynomial 6y² + 30y is 6y. Extracting this GCF simplifies the problem and reveals the structure of the polynomial.

Prime Polynomials

After factoring out the GCF, it’s important to identify if the resulting polynomial can be factored further or if it's a prime polynomial.

A **prime polynomial** is a polynomial that cannot be factored into the product of two non-trivial polynomials. For example, consider the polynomial y + 5. It cannot be broken down further into simpler polynomials, making it prime.
Prime polynomials play a crucial role because they simplify the overall polynomial expression and sometimes serve as the final step in the factoring process.

Algebraic Expressions

Understanding polynomials requires familiarity with algebraic expressions. An algebraic expression is a mathematical phrase that includes numbers, variables (like x or y), and operation symbols.

**Types of Terms in Polynomials**:
* **Constant Term**: A number without any variables, e.g., 5.
* **Linear Term**: A term with a variable to the first power, e.g., 6y.
* **Quadratic Term**: A term with a variable to the second power, e.g., 6y².
**Operations on Polynomials**:
* **Addition and Subtraction**: Combine like terms (terms with the same variable part).
* **Multiplication**: Use distributive property to expand the product.
* **Factoring**: Express the polynomial as a product of simpler polynomials. This includes factoring out the GCF.
Having a strong grasp of these elements will help you confidently tackle polynomial problems.