Step-by-step explanation:
this just means that the main expression for the volume is the result of the multiplication of 3 terms - each representing one dimension of the 3 to define a volume (length, width, height).
how do we find out, which one is such a factor ?
well, each factor or multiplication term defines a zero point of the main expression.
we need to find fit which x value these terms turn 0, and then we try this x value in the main expression, and see if it also turns 0. if yes, we have found a term representing a dimension.
3x + 6 = 0 when x = -2
3(-2)³ - 18(-2)² + 4(-2) - 24 =
= -24 - 72 - 8 - 24 = -128
not 0, therefore not a factor (or dimension).
3x - 6 = 0 when x = 2
3×2³ - 18×2² + 4×2 - 24 =
= 24 - 72 + 8 - 24 = -64
not 0, therefore not a factor (or dimension).
x - 4 = 0 when x = 4
3×4³ - 18×4² + 4×4 - 24 =
= 192 - 288 + 16 - 24 = -104
not 0, therefore not a factor (or dimension).
3x + 4 = 0 when x = -4/3
3×(-4/3)³ - 18×(-4/3)² + 4×-4/3 - 24 =
= -192/27 - 288/9 - 16/3 - 24 =
= -64/3 - 96/3 - 16/3 - 24 = -176/3 - 24 =
= -176/3 - 72/3 = -248/3
not 0, therefore not a factor (or dimension).
none of these 4 terms are a factor or dimension.
but (x - 6) would be one
x - 6 = 0 when x = 6
3×6³ - 18×6² + 4×6 - 24 =
= 648 - 648 + 24 - 24 = 0