Certainly! Let's work through the given polynomial step by step to arrive at a detailed solution.
We are given:
[tex]\[ p = -5 \][/tex]
[tex]\[ q = -2 \][/tex]
We need to form the polynomial with these values:
[tex]\[ x^3 + p x^2 + q x + 1 \][/tex]
First, substitute the given values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into the polynomial:
[tex]\[ x^3 + (-5)x^2 + (-2)x + 1 \][/tex]
This simplifies to:
[tex]\[ x^3 - 5x^2 - 2x + 1 \][/tex]
We can represent this polynomial visually and mathematically as:
[tex]\[ \text{Polynomial} = x^3 - 5x^2 - 2x + 1 \][/tex]
Thus, the polynomial is:
[tex]\[ x^3 - 5x^2 - 2x + 1 \][/tex]
In a more formal mathematical notation, this polynomial can be expressed as a polynomial in [tex]\( x \)[/tex] with integer coefficients and parameters [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
[tex]\[ \text{Poly}(x^3 + p x^2 + q x + 1, x, \text{domain} = \mathbb{Z}[p, q]) \][/tex]
Substituting [tex]\( p = -5 \)[/tex] and [tex]\( q = -2 \)[/tex] into this expression, we get:
[tex]\[ \text{Poly}(x^3 - 5x^2 - 2x + 1, x, \text{domain} = \mathbb{Z}[p, q]) \][/tex]
In summary, the polynomial formed with the given values [tex]\( p = -5 \)[/tex] and [tex]\( q = -2 \)[/tex] is:
[tex]\[ \boxed{x^3 - 5x^2 - 2x + 1} \][/tex]
