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Let's go through each question one at a time.

### Question 1
To find the remainder when \(4x^3 - 5x + 1\) is divided by each of the expressions, we can use the **Remainder Theorem**. According to this theorem, if we want to find the remainder of a polynomial \(f(x)\) when divided by \(x - a\), we evaluate \(f(a)\).

Given \(f(x) = 4x^3 - 5x + 1\):

#### a) \( x - 2 \)
To find the remainder when \(f(x)\) is divided by \(x - 2\), we substitute \(x = 2\):
\[
f(2) = 4(2)^3 - 5(2) + 1 = 4 \cdot 8 - 10 + 1 = 32 - 10 + 1 = 23
\]
So, the remainder is **23**.

#### b) \( x + 3 \)
To find the remainder when \(f(x)\) is divided by \(x + 3\), we substitute \(x = -3\):
\[
f(-3) = 4(-3)^3 - 5(-3) + 1 = 4 \cdot (-27) + 15 + 1 = -108 + 15 + 1 = -92
\]
The remainder is **-92**.

#### c) \( 2x - 1 \)
When dividing by \(2x - 1\), we substitute \(x = \frac{1}{2}\):
\[
f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^3 - 5\left(\frac{1}{2}\right) + 1 = 4 \cdot \frac{1}{8} - \frac{5}{2} + 1 = \frac{1}{2} - \frac{5}{2} + 1 = -1
\]
The remainder is **-1**.

### Question 2
We need to determine the value of \(p\) such that the expression \(4x^2 - px + 7\) leaves a remainder of \(-2\) when divided by \(x - 3\).

Using the Remainder Theorem, we substitute \(x = 3\) and set the result equal to \(-2\):
\[
f(3) = 4(3)^2 - p(3) + 7 = -2
\]
\[
4 \cdot 9 - 3p + 7 = -2
\]
\[
36 - 3p + 7 = -2
\]
\[
43 - 3p = -2
\]
\[
-3p = -45
\]
\[
p = 15
\]
So, the value of \(p\) is **15**.

Would you like more details or have any further questions?

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**Related Questions:**
1. How does the Remainder Theorem apply to polynomials of higher degrees?
2. What is the difference between the Remainder Theorem and Synthetic Division?
3. How would we find the quotient in addition to the remainder?
4. What changes when dividing by a binomial like \(2x - 1\) instead of \(x - a\)?
5. How would we determine the remainder if the divisor was \(x^2 - 4\)?

**Tip:** When using the Remainder Theorem, always check if the divisor is of the form \(x - a\) for straightforward substitution.

1. Find the remainder when 4x^3 - 5x + 1 is divided by: a) x - 2 b) x + 3 c) 2x - 1

2. The expression 4x^2 - px + 7 leaves a remainder of -2 when divided by x - 3. Find the value of p.

Learn how to find the remainder when dividing the polynomial 4x^3 - 5x + 1 by x - 2, x + 3, and 2x - 1. Also, find the value of p when the polynomial 4x^2 - px + 7 leaves a remainder of -2 when divided by x - 3. This problem covers key algebraic concepts including the Remainder Theorem.

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