Problem 19 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 3 y^{2}-6 $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial $$3y^2 - 6$$ is $$3(y^2 - 2)$$.

Step 1: Identify Common Factors

The given polynomial is $$3y^2 - 6$$. Let's look for any common factors in both terms. We can see that the number 3 is a common factor in both $$3y^2$$ and $$-6$$.

Step 2: Factor out common factor

Now that we have identified the common factor 3, we can factor it out from the polynomial. Factoring out 3 from both terms, we get: $$3(y^2 - 2)$$

Step 3: Check for further factorization

Now we have the factored form $$3(y^2 - 2)$$. The remaining expression inside the parentheses, $$(y^2 - 2)$$, cannot be factored any further as it is a difference of two squares but the 2 is not a perfect square.

Step 4: Write the final factored form

We have factored the given polynomial completely: $$3y^2 - 6 = 3(y^2 - 2)$$

Key Concepts

Common Factors

Understanding common factors is essential when working with algebraic expressions. A common factor refers to a number or variable that is shared by each term within a polynomial. For example, given the polynomial \(3y^2 - 6\), we can observe that the number 3 is a factor of both terms, \(3y^2\) and \(-6\).

Identifying these shared factors allows us to simplify expressions by factoring them out. When we factor out the common factor from each term, we’re effectively dividing each term by this factor. The factored form will include the common factor multiplied by a simpler expression, which in the case of our original exercise resulst in \(3(y^2 - 2)\). This step is the cornerstone of factorization and is crucial for further simplification of algebraic expressions.

Difference of Squares

The difference of squares is a special pattern in elementary algebra that emerges when subtracting one square number from another. This pattern can be factored into the product of a sum and difference of the square roots of the original terms. The generic formula for this is \(a^2 - b^2 = (a + b)(a - b)\).

This is particularly useful when dealing with polynomials. However, the factoring is only possible if both terms are perfect squares. In the exercise \(3(y^2 - 2)\), the polynomial inside the parentheses \((y^2 - 2)\) represents a difference of two terms, but it isn't a difference of squares because 2 is not a perfect square. Understanding this concept prevents the unnecessary search for further factorization where it is not applicable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (e.g., addition, subtraction, multiplication, division). In the context of our exercise, \(3y^2 - 6\) is an algebraic expression with variables (\(y\)), coefficients (3), and constants (-6) combined through subtraction and multiplication.

At the core of working with algebraic expressions is the need to manipulate and simplify them. Factoring is a form of simplification where expressions are re-written as the product of their factors. It is a fundamental skill in elementary algebra that aids in solving equations, finding zeros of functions, and integrating more complex calculus operations.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the manipulation and simplification of algebraic expressions and equations. It covers basic operations and their properties, the use of variables, and techniques such as factoring.

In the provided exercise, several elementary algebra techniques are employed. Identifying common factors, understanding when to stop factoring, and recording the final simplified form of an algebraic expression are all part of the problem-solving toolkit in elementary algebra. This discipline lays the groundwork for more advanced studies in mathematics, as it builds the intuitive understanding of how numbers and variables can interact and be transformed.