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To determine the first three terms in the expansion of \((1 - 2x^5)(1 + x)^7\) in ascending powers of \(x\), we can expand each part and then find the product of the relevant terms to get the desired powers of \(x\).

1. **Expand \((1 + x)^7\)** using the Binomial Theorem:
   \[
   (1 + x)^7 = \sum_{k=0}^7 \binom{7}{k} x^k = 1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7
   \]

2. **Consider \((1 - 2x^5)\)**:
   \[
   1 - 2x^5
   \]
   Since the \(x^5\) term in \((1 - 2x^5)\) will only impact the terms of the expansion of \((1 + x)^7\) from the fifth term onwards, it is straightforward to identify the interactions:
   \[
   (1 - 2x^5)(1 + x)^7 = (1 + x)^7 - 2x^5(1 + x)^7
   \]
   To get the first three terms in ascending powers of \(x\), we focus on the expansions that contribute to terms of order less than \(x^5\).

3. **Calculate the product and simplify**:
   \[
   \begin{align*}
   (1 - 2x^5)(1 + x)^7 &= (1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7) - 2x^5(1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7) \\
   &= 1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5(1 - 2) + 7x^6(-2) + x^7(-2) \\
   &= 1 + 7x + 21x^2 + 35x^3 + 35x^4 - 21x^5 - 14x^6 - 2x^7.
   \end{align*}
   \]
   Focusing on the terms up to \(x^3\):
   \[
   1 + 7x + 21x^2 + 35x^3.
   \]
   
The first three terms in the expansion in ascending powers of \(x\) are \(1\), \(7x\), and \(21x^2\). Would you like more details or have any other questions about this expansion?

### Additional Questions
1. What happens if we expand further to find terms including and beyond \(x^4\)?
2. How does the binomial theorem help in quickly expanding powers like \((1 + x)^7\)?
3. Can the negative coefficient \(-2x^5\) in \((1 - 2x^5)\) affect terms of lower degree like \(x^3\) or \(x^4\)?
4. How can we use a different method, like a computer algebra system, to verify this expansion?
5. What is the importance of recognizing patterns in the coefficients of binomial expansions?

### Tip
To easily remember binomial coefficients for small powers, you can reference Pascal's Triangle, where each number is the sum of the two numbers directly above it.

Determine the first three terms in the expansion of (1 - 2x)^5 (1 + x)^7 in ascending powers of x.

Learn how to find the first three terms in the expansion of (1 - 2x)^5 (1 + x)^7 using binomial expansion and polynomial multiplication. This problem covers essential algebraic concepts and is suitable for high school students.

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