The problem you uploaded describes a polynomial function and involves two conditions for determining the values of constants $p$ and $q$. Specifically, it mentions that the polynomial function:

$f(x) = px^3 - x^2 + qx - 2$

has no remainder when divided by $x - 1$, and the remainder when divided by $x + 2$ is -18.

Let's break it down:

1. No remainder when divided by $x - 1$: This means that $f(1) = 0$, because a polynomial with no remainder when divided by $x - 1$ implies that $x = 1$ is a root of the polynomial.

2. Remainder of -18 when divided by $x + 2$: This means that $f(-2) = -18$, because the remainder theorem tells us that the remainder when a polynomial is divided by $x + 2$ is simply $f(-2)$.

To solve for $p$ and $q$, we'll use these two conditions:

1. $f(1) = 0$ to get the first equation.
2. $f(-2) = -18$ to get the second equation.

Now, I will calculate these values and set up the system of equations to solve for $p$ and $q$.

Let me go ahead and compute that. sympy import symbols, Eq, solve

Define the variables p and q

p, q = symbols('p q')

Define the polynomial function

f = p * (1)**3 - (1)**2 + q * (1) - 2

First condition: f(1) = 0

equation1 = Eq(f, 0)

Define the second equation f(-2) = -18

f2 = p * (-2)**3 - (-2)**2 + q * (-2) - 2

Second condition: f(-2) = -18

equation2 = Eq(f2, -18)

Solve the system of equations

solution = solve((equation1, equation2), (p, q)) solutionThe values of the constants $p$ and $q$ are:

$p = 1 \quad \text{and} \quad q = 2$

This satisfies the given conditions for the polynomial function.

If you need any further explanation or additional steps, feel free to ask! Here are some related questions for you to explore further:

1. How would the polynomial change if $p$ were a different value?
2. How can you use synthetic division to verify the remainder of a polynomial division?
3. What is the Remainder Theorem, and how does it apply in polynomial division?
4. How do you factor a cubic polynomial using the roots and synthetic division?
5. What other methods can you use to solve a system of equations involving polynomials?

Tip: Always check that your equations align with the conditions given in the problem—substituting values into your equation is a reliable verification method.