https://file.aiquickdraw.com/mathbot/file/172740539641836et2xck.jpg

Let's analyze and solve the two problems visible in the image:

### Problem 11:
\[
y = \frac{7}{x^3} - \frac{5}{x}
\]

First, simplify the terms:
\[
y = 7x^{-3} - 5x^{-1}
\]

Next, let's differentiate this function with respect to \( x \):
\[
\frac{dy}{dx} = \frac{d}{dx}\left(7x^{-3} - 5x^{-1}\right)
\]

Using the power rule \(\frac{d}{dx}(x^n) = n x^{n-1}\):
\[
\frac{dy}{dx} = 7(-3)x^{-4} - 5(-1)x^{-2}
\]
\[
\frac{dy}{dx} = -21x^{-4} + 5x^{-2}
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = -\frac{21}{x^4} + \frac{5}{x^2}
\]

### Problem 13:
\[
y = (2x^5 - 8x^3 - 7x - 1)(11x^3 - x^2 + 1)
\]

To solve this, we will expand the product of the two polynomials using the distributive property. I'll proceed step by step.

1. Multiply each term from the first polynomial with each term from the second polynomial:
\[
y = (2x^5)(11x^3 - x^2 + 1) + (-8x^3)(11x^3 - x^2 + 1) + (-7x)(11x^3 - x^2 + 1) + (-1)(11x^3 - x^2 + 1)
\]

Expanding:
\[
y = 2x^5(11x^3) + 2x^5(-x^2) + 2x^5(1)
\]
\[
y = -8x^3(11x^3) - 8x^3(-x^2) - 8x^3(1)
\]
\[
y = -7x(11x^3) - 7x(-x^2) - 7x(1)
\]
\[
y = -1(11x^3) - 1(-x^2) - 1(1)
\]

After multiplying the terms:
\[
y = 22x^8 - 2x^7 + 2x^5 - 88x^6 + 8x^5 - 8x^3 - 77x^4 + 7x^3 - 7x - 11x^3 + x^2 - 1
\]

Now, combine like terms:
\[
y = 22x^8 - 2x^7 - 88x^6 + 10x^5 - 77x^4 - 12x^3 + x^2 - 7x - 1
\]

This is the fully expanded form of the polynomial.

---

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How would you solve for \( x \) when the derivative in problem 11 equals zero?
2. Can you find the second derivative for problem 11?
3. What is the degree of the expanded polynomial in problem 13?
4. How would you factor the result from problem 13?
5. Can you find critical points of the polynomial from problem 13?

**Tip:** When expanding polynomials, organizing the terms by degree can help combine like terms efficiently.

y = (2x^5 - 8x^3 - 7x - 1)(11x^3 - x^2 + 1), Find the expanded form of the product of the two polynomials.

Learn how to expand the product of two polynomials: (2x^5 - 8x^3 - 7x - 1) and (11x^3 - x^2 + 1). Detailed step-by-step solution covers key algebraic and polynomial multiplication concepts. Suitable for students in Grades 11-12.

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