Sure, let's factor the given expression step-by-step:
[tex]\[
3x^3 - 12xy^2
\][/tex]
1. Identify Common Factors:
Notice that each term in the expression has a common factor of [tex]\(3x\)[/tex]:
[tex]\[
3x^3 - 12xy^2 = 3x(x^2 - 4y^2)
\][/tex]
2. Factor the Quadratic Expression:
Next, we need to factor the quadratic expression inside the parentheses, [tex]\(x^2 - 4y^2\)[/tex]. This is a difference of squares, which can be factored using the difference of squares formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
x^2 - 4y^2 = (x - 2y)(x + 2y)
\][/tex]
3. Combine All Factors:
Now, combine the common factor [tex]\(3x\)[/tex] with the factored quadratic expression:
[tex]\[
3x(x^2 - 4y^2) = 3x(x - 2y)(x + 2y)
\][/tex]
Therefore, the factorised form of the given expression [tex]\(3x^3 - 12xy^2\)[/tex] is:
[tex]\[
3x(x - 2y)(x + 2y)
\][/tex]
