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This question involves expanding the binomial expression \((1 - 2x)^p\) and using the binomial theorem to find specific coefficients and the value of the constant \(p\). The binomial expansion is given by:

\[
(1 - 2x)^p = \sum_{k=0}^{p} \binom{p}{k} (-2x)^k
\]

Let's break this down:

### Part (a): Find the value of \(p\)
We are given that the coefficient of \(x^2\) is 40. In the expansion, the general term for \(x^k\) is:

\[
T_k = \binom{p}{k} (1)^{p-k} (-2x)^k = \binom{p}{k} (-2)^k x^k
\]

For \(x^2\), we have:

\[
T_2 = \binom{p}{2} (-2)^2 = \binom{p}{2} \cdot 4
\]

This coefficient is equal to 40, so:

\[
\binom{p}{2} \cdot 4 = 40
\]

Dividing both sides by 4:

\[
\binom{p}{2} = 10
\]

Now, recall the binomial coefficient formula:

\[
\binom{p}{2} = \frac{p(p-1)}{2}
\]

So:

\[
\frac{p(p-1)}{2} = 10
\]

Multiplying both sides by 2:

\[
p(p-1) = 20
\]

Solving this quadratic equation:

\[
p^2 - p - 20 = 0
\]

Using the quadratic formula:

\[
p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)} = \frac{1 \pm \sqrt{1 + 80}}{2} = \frac{1 \pm \sqrt{81}}{2} = \frac{1 \pm 9}{2}
\]

Thus, \(p = \frac{1 + 9}{2} = 5\) or \(p = \frac{1 - 9}{2} = -4\).

Since \(p > 0\), we have \(p = 5\).

### Part (b): Coefficient of \(x\)
The general term is:

\[
T_1 = \binom{p}{1} (-2)^1 x = \binom{5}{1} (-2) = 5(-2) = -10
\]

So, the coefficient of \(x\) is \(-10\).

### Part (c): Coefficient of \(x^3\)
For \(x^3\), the term is:

\[
T_3 = \binom{p}{3} (-2)^3 = \binom{5}{3} (-8)
\]

Calculating \(\binom{5}{3}\):

\[
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
\]

Thus, the coefficient of \(x^3\) is:

\[
10(-8) = -80
\]

### Final answers:
- (a) \(p = 5\)
- (b) Coefficient of \(x = -10\)
- (c) Coefficient of \(x^3 = -80\)

Would you like more details or have any further questions? Here are some related problems:

1. How would the answer change if the coefficient of \(x^2\) was different?
2. What would be the coefficient of \(x^4\) in this expansion?
3. How does the binomial theorem apply if the power \(p\) is negative?
4. Can you apply this method for fractional values of \(p\)?
5. How can you verify these coefficients without expanding the entire binomial?

**Tip**: The binomial theorem is a powerful tool for expanding expressions of the form \((a + b)^n\). Mastering it helps with quick calculations of specific terms in polynomial expansions.

When (1 − 2x)^p is expanded, the coefficient of x^2 is 40. Given that p > 0, find the following:
a) the value of the constant p
b) the coefficient of x
c) the coefficient of x^3

Learn how to solve a binomial expansion problem where the coefficient of x^2 is 40. Discover how to find the constant p, as well as the coefficients of x and x^3 using the binomial theorem. Suitable for advanced high school students.

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