Problem 7 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 6 y^{2}-24 $$

Step-by-Step Solution

Answer

The fully factored form of the expression \( 6y^{2}-24 \) is \( 6(y + 2)(y - 2) \)

Step 1: Identify Common Factors

The first step is to identify any common factors that can be pulled out from each term in the expression. Looking at both terms, \( 6y^{2} \) and \( -24 \), it's clear that they both share a common factor of 6.

Step 2: Factor Out the Greatest Common Factor

This involves rewriting the original expression as the product of this greatest common factor and another polynomial. Factoring out the common factor of 6 from each term leaves us with \( 6(y^{2} - 4) \).

Step 3: Factor the Resulting Polynomial Further

At this point, the polynomial \( y^{2} - 4 \) can be factored further because it is a difference of squares. A difference of squares is a characteristic pattern that can be rewritten as the product of the sum and the difference of two terms. In this case, \( y^{2} - 4 \) could be rewritten as \( (y + 2)(y - 2) \).

Step 4: Write the Final Factored Form

The final step is to substitute \( (y + 2)(y - 2) \) back into the expression from Step 2 to get the final factored form of our expression. This results in \( 6(y + 2)(y - 2) \).