Introduction to Factorization:

Factorization is a fundamental algebraic skill in mathematics, which involves expressing a polynomial as a product of its factors. For the expression 8y^3 - 6y, we will explore the process of fully factorizing it step by step.

Understanding the Expression

The given expression is 8y^3 - 6y. Before we start the factorization process, it's important to understand the expression.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in the factorization process is to identify the greatest common factor of the coefficients. Here, the coefficients are 8 and 6.

The prime factorization of 8 is 2^3. The prime factorization of 6 is 2 * 3.

The greatest common factor of 8 and 6 is 2. Additionally, both terms contain y. Therefore, the greatest common factor (GCF) of the terms 8y^3 - 6y is 2y.

Step 2: Factor Out the GCF

Next, we factor out the GCF from the given expression.

8y^3 - 6y 2y(4y^2 - 3)

By factoring out 2y, we get:

2y(4y^2 - 3)

Now, the expression inside the parentheses, 4y^2 - 3, needs to be checked to see if it can be factored further.

Step 3: Check for Further Factorization

The expression 4y^2 - 3 is a difference of squares. The difference of squares formula is:

a^2 - b^2 (a - b)(a b)

In this case, a 2y and b sqrt{3}. Therefore, we can rewrite the expression as:

(2y - sqrt{3})(2y sqrt{3})

So, the fully factored form of the expression 8y^3 - 6y is:

2y(2y - sqrt{3})(2y sqrt{3})

Conclusion

Thus, the fully factored form of the expression 8y^3 - 6y is:

2y(2y^2 - 3)

or more precisely, when further simplified:

2y(2y - sqrt{3})(2y sqrt{3})

Additional Queries

What factor is common to 8 and 6? The factor is 2. What factor is common to y^3 and y? The factor is y. Then what is the Greatest Common Factor that can be pulled out of the two terms? It is 2y. What is left after you divide 8y^3 by 2y? The result is 4y^2. What is left after you divide -6y by 2y? The result is -3.

By following these steps, we can fully factorize the given expression accurately.