Answer :

We are given the polynomial

[tex]$$
P(x) = 2x^3 - 5x^2 + 25.
$$[/tex]

Below is a step-by-step explanation of how to obtain its factored form and its derivative.

1. Original Polynomial:
The polynomial is already given as

[tex]$$
P(x) = 2x^3 - 5x^2 + 25.
$$[/tex]

2. Factoring the Polynomial:
When attempting to factor [tex]$P(x)$[/tex] over the integers or the rationals, we find that it does not factor into simpler polynomial terms. Therefore, the factored form remains

[tex]$$
2x^3 - 5x^2 + 25.
$$[/tex]

3. Differentiating the Polynomial:
To compute the derivative [tex]$P'(x)$[/tex], differentiate each term of [tex]$P(x)$[/tex] with respect to [tex]$x$[/tex]:

- The derivative of [tex]$2x^3$[/tex] is

[tex]$$
\frac{d}{dx}(2x^3) = 6x^2.
$$[/tex]

- The derivative of [tex]$-5x^2$[/tex] is

[tex]$$
\frac{d}{dx}(-5x^2) = -10x.
$$[/tex]

- The derivative of the constant [tex]$25$[/tex] is

[tex]$$
\frac{d}{dx}(25) = 0.
$$[/tex]

Combining these, the derivative of the polynomial is

[tex]$$
P'(x) = 6x^2 - 10x.
$$[/tex]

Final Answer:

- The original polynomial is

[tex]$$
2x^3 - 5x^2 + 25.
$$[/tex]

- Its factored form is

[tex]$$
2x^3 - 5x^2 + 25.
$$[/tex]

- Its derivative is

[tex]$$
6x^2 - 10x.
$$[/tex]

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