Question

determine the product of three linear factors. verify graphically that the expressions are equivalent.
sample problem
$3x(x + 3)(x - 2)$
$= 3x(x^2 - 2x + 3x - 6)$
$= 3x(x^2 + x - 6)$
$= 3x^3 + 3x^2 - 18x$
the graph of the original expression and the graph of the final expression are the same. so the expressions are equivalent.
$0.25x(12x - 1)(8 - 3x)$
$=\boxed{}$
type the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Multiply the two binomials

First, expand $(12x - 1)(8 - 3x)$ using the FOIL method:
$$(12x)(8) + (12x)(-3x) + (-1)(8) + (-1)(-3x) = 96x - 36x^2 - 8 + 3x$$
Combine like terms:
$$-36x^2 + (96x + 3x) - 8 = -36x^2 + 99x - 8$$

Step2: Multiply by $0.25x$

Distribute $0.25x$ across the quadratic expression:
$$0.25x(-36x^2 + 99x - 8) = 0.25x(-36x^2) + 0.25x(99x) + 0.25x(-8)$$
Calculate each term:
$$= -9x^3 + \frac{99}{4}x^2 - 2x = -9x^3 + 24.75x^2 - 2x$$

Answer:

$-9x^3 + 24.75x^2 - 2x$

Graphical verification: When plotting both the original factored expression $0.25x(12x - 1)(8 - 3x)$ and the expanded polynomial $-9x^3 + 24.75x^2 - 2x$, the two graphs will be identical, confirming they are equivalent.