Let's solve these polynomial factorization problems one by one.

36) $x^2 - 6xy + 9y^2$

This is a quadratic trinomial. It resembles a perfect square trinomial:

$x^2 - 6xy + 9y^2 = (x - 3y)^2$

Thus, the factored form is:

$(x - 3y)^2$

37) $9x^2 - a^2x^3 - 9x^2 + a^2$

First, let's simplify:

$9x^2 - 9x^2 + a^2 - a^2x^3 = a^2(1 - x^3)$

Factoring $1 - x^3$ using the difference of cubes:

$1 - x^3 = (1 - x)(1 + x + x^2)$

Thus, the factored form of the entire expression is:

$a^2(1 - x)(1 + x + x^2)$

38) $3xy - 5rx - 10rs + 6sy$

Group the terms to factor by grouping:

$(3xy + 6sy) - (5rx + 10rs)$

Factor out common terms in each group:

$3y(x + 2s) - 5r(x + 2s)$

Now factor the common binomial $x + 2s$:

$(3y - 5r)(x + 2s)$

39) $a^2b^2 - 25c^2$

This is a difference of squares:

$a^2b^2 - 25c^2 = (ab - 5c)(ab + 5c)$

40) $6x^2 + 13xy - 5y^2$

We need to factor this trinomial. We look for two numbers that multiply to $6 \times -5 = -30$ and add to $13$. These numbers are $15$ and $-2$. Thus, rewrite the middle term:

$6x^2 + 15xy - 2xy - 5y^2$

Group terms:

$(6x^2 + 15xy) - (2xy + 5y^2)$

Factor each group:

$3x(2x + 5y) - y(2x + 5y)$

Now factor the common binomial $2x + 5y$:

$(3x - y)(2x + 5y)$

Summary of Solutions:

1. $x^2 - 6xy + 9y^2 = (x - 3y)^2$
2. $9x^2 - a^2x^3 - 9x^2 + a^2 = a^2(1 - x)(1 + x + x^2)$
3. $3xy - 5rx - 10rs + 6sy = (3y - 5r)(x + 2s)$
4. $a^2b^2 - 25c^2 = (ab - 5c)(ab + 5c)$
5. $6x^2 + 13xy - 5y^2 = (3x - y)(2x + 5y)$

Would you like further details on any of these steps?

5 Related Questions:

1. How can you recognize a perfect square trinomial?
2. What is the difference between factoring by grouping and factoring a difference of squares?
3. How do you factor higher-degree polynomials using special identities like the sum or difference of cubes?
4. Why is factoring important in simplifying rational expressions?
5. What is the general process for factoring trinomials of the form $ax^2 + bx + c$?

Tip:

Always check your factored answer by expanding it back out to ensure it matches the original expression!