Question

factor $12y^{3}-3y$ completely.

Explanation:

Step1: Find the GCF

Find the greatest common factor (GCF) of \(12y^3\) and \(3y\). The GCF of 12 and 3 is 3, and the GCF of \(y^3\) and \(y\) is \(y\). So the GCF is \(3y\).
\[12y^3 - 3y = 3y(4y^2 - 1)\]

Step2: Factor the difference of squares

Notice that \(4y^2 - 1\) is a difference of squares, since \(4y^2=(2y)^2\) and \(1 = 1^2\). The formula for factoring a difference of squares is \(a^2 - b^2=(a + b)(a - b)\). Here, \(a = 2y\) and \(b = 1\).
\[4y^2 - 1=(2y + 1)(2y - 1)\]

Step3: Combine the factors

Substitute the factored form of \(4y^2 - 1\) back into the expression from Step 1.
\[12y^3 - 3y=3y(2y + 1)(2y - 1)\]

Answer:

\(3y(2y + 1)(2y - 1)\)