Question

which expressions are factors of $3x^3 - 48x$?
i. $(x - 4)$
ii. $(x + 4)$
iii. $(x - 3)$
iv. $(x + 3)$
v. $(3x)$
a) i, ii, v
b) ii, iv, v
c) iii, iv, v
d) i, iii, v

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from \(3x^3 - 48x\). The GCF of \(3x^3\) and \(48x\) is \(3x\). So, we have:
\(3x^3 - 48x = 3x(x^2 - 16)\)

Step2: Factor the difference of squares

Notice that \(x^2 - 16\) is a difference of squares, which factors as \((x - 4)(x + 4)\) (since \(a^2 - b^2=(a - b)(a + b)\) with \(a = x\) and \(b = 4\)). So now the expression is:
\(3x(x - 4)(x + 4)\)

Step3: Identify the factors

From the factored form \(3x(x - 4)(x + 4)\), the factors are \(3x\), \((x - 4)\), and \((x + 4)\). So the expressions that are factors are I. \((x - 4)\), II. \((x + 4)\), and V. \((3x)\).

Answer:

A. I, II, V