https://file.aiquickdraw.com/mathbot/file/1726004120104q7k43s8o.jpg

Let's break down and solve the problems in the image one by one.

### Problem 19: \( 8m^3 - 25 \)
This expression cannot be factored further, as it is not a difference of cubes or squares, and there is no common factor.

Answer: The expression \( 8m^3 - 25 \) is already simplified.

---

### Problem 20: \( 2t^3 + 32t^2 + 128t \)
This expression has a common factor of \( 2t \) in all terms:

\[
2t(t^2 + 16t + 64)
\]

Now, factor the quadratic:

\[
t^2 + 16t + 64 = (t + 8)^2
\]

Thus, the fully factored form is:

\[
2t(t + 8)^2
\]

Answer: \( 2t(t + 8)^2 \)

---

### Problem 21: \( 5y^5 + 135y^2 \)
First, factor out the greatest common factor, which is \( 5y^2 \):

\[
5y^2(y^3 + 27)
\]

Now, recognize that \( y^3 + 27 \) is a sum of cubes:

\[
y^3 + 27 = (y + 3)(y^2 - 3y + 9)
\]

Thus, the fully factored form is:

\[
5y^2(y + 3)(y^2 - 3y + 9)
\]

Answer: \( 5y^2(y + 3)(y^2 - 3y + 9) \)

---

### Problem 22: \( 81x^4 - 16 \)
This is a difference of squares:

\[
81x^4 - 16 = (9x^2 - 4)(9x^2 + 4)
\]

The term \( 9x^2 - 4 \) is also a difference of squares:

\[
9x^2 - 4 = (3x - 2)(3x + 2)
\]

Thus, the fully factored form is:

\[
(3x - 2)(3x + 2)(9x^2 + 4)
\]

Answer: \( (3x - 2)(3x + 2)(9x^2 + 4) \)

---

### Problem 23: \( \frac{x^2 - 16}{x^2 + x - 20} \)
First, factor the numerator and denominator.

Numerator \( x^2 - 16 \) is a difference of squares:

\[
x^2 - 16 = (x - 4)(x + 4)
\]

Now, factor the denominator \( x^2 + x - 20 \). Look for two numbers that multiply to \( -20 \) and add to \( 1 \):

\[
x^2 + x - 20 = (x - 4)(x + 5)
\]

Now simplify the fraction:

\[
\frac{(x - 4)(x + 4)}{(x - 4)(x + 5)} = \frac{x + 4}{x + 5} \quad (\text{for } x \neq 4)
\]

Answer: \( \frac{x + 4}{x + 5} \)

---

Would you like to go over the remaining problems (24 and 25) or any specific steps in more detail?

Here are 5 related questions for further exploration:
1. How can the sum of cubes formula be applied in more complex factoring problems?
2. What is the general method for factoring a quadratic trinomial?
3. How does the difference of squares factorization work in higher-degree polynomials?
4. When simplifying rational expressions, why is it important to check for excluded values?
5. What are the different types of special products that frequently appear in factoring?

**Tip**: Always check for common factors first when factoring any polynomial. This simplifies the process and reduces the number of terms to deal with.

Solve and simplify the given algebraic expressions from the image.

Learn how to factor and simplify algebraic expressions involving the difference of squares, sum of cubes, and rational expressions. Explore step-by-step solutions for problems like 8m^3 - 25 and x^2 - 16/x^2 + x - 20. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation