https://file.aiquickdraw.com/mathbot/file/1728747065309um449t0c.jpg

Let's break down the three expressions for factorization from the image:

### a) \( 24x^4 - 54 \)

This is a difference of two terms. We start by factoring out the greatest common factor (GCF).

- The GCF of 24 and 54 is 6, so:
  \[
  24x^4 - 54 = 6(4x^4 - 9)
  \]
- Next, \( 4x^4 - 9 \) is a difference of squares:
  \[
  4x^4 - 9 = (2x^2 - 3)(2x^2 + 3)
  \]
- So the fully factored expression is:
  \[
  24x^4 - 54 = 6(2x^2 - 3)(2x^2 + 3)
  \]

### b) \( 32x^4 - 98y^4 \)

Again, we begin by factoring out the GCF.

- The GCF of 32 and 98 is 2, so:
  \[
  32x^4 - 98y^4 = 2(16x^4 - 49y^4)
  \]
- Now factor \( 16x^4 - 49y^4 \), which is also a difference of squares:
  \[
  16x^4 - 49y^4 = (4x^2 - 7y^2)(4x^2 + 7y^2)
  \]
- So the fully factored expression is:
  \[
  32x^4 - 98y^4 = 2(4x^2 - 7y^2)(4x^2 + 7y^2)
  \]

### c) \( (2x - 5)^2 - 16 \)

This is another difference of squares.

- The difference of squares formula is \( a^2 - b^2 = (a - b)(a + b) \), where \( a = (2x - 5) \) and \( b = 4 \) (since \( 16 = 4^2 \)).
- Applying the formula:
  \[
  (2x - 5)^2 - 16 = [(2x - 5) - 4][(2x - 5) + 4]
  \]
  Simplifying both terms:
  \[
  (2x - 9)(2x - 1)
  \]
- So the fully factored expression is:
  \[
  (2x - 5)^2 - 16 = (2x - 9)(2x - 1)
  \]

Would you like further details on any of these steps? Here are five related questions to consider:

1. How do you factor a difference of cubes?
2. What is the difference between factoring quadratics and quartics?
3. How can you recognize a difference of squares quickly?
4. Why is factoring important in solving polynomial equations?
5. Can all polynomials be factored over real numbers?

**Tip**: Always check for a greatest common factor (GCF) before trying other factorization techniques.

Factorise the following expressions completely: a) 24x^4 - 54, b) 32x^4 - 98y^4, c) (2x - 5)^2 - 16

Learn how to factor polynomials completely with examples including difference of squares and greatest common factor. This problem covers factorizations of expressions like 24x^4 - 54, 32x^4 - 98y^4, and (2x - 5)^2 - 16, suitable for Grades 10-12.

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