Question

factor the expression completely. $3x^3 - 10x$

Explanation:

Step1: Identify the GCF

The greatest common factor (GCF) of \(3x^3\) and \(-10x\) is \(x\) since \(x\) is the highest power of \(x\) that divides both terms and there is no common numerical factor other than 1 (since 3 and 10 are coprime).
\[
3x^3 - 10x = x(3x^2 - 10)
\]

Step2: Check for further factoring

The quadratic \(3x^2 - 10\) is a difference of squares? No, because \(10\) is not a perfect square. Also, it has no real - valued linear factors with integer coefficients (the discriminant of \(ax^2+bx + c\) is \(b^2 - 4ac=0 - 4\times3\times(- 10)=120\), which is not a perfect square, so it doesn't factor over the integers). So the factoring is complete.

Answer:

\(x(3x^{2}-10)\)