Factoring Polynomials Worksheet - Practice problems for algebra students.

It looks like you've uploaded a worksheet with two sections: Factorise (I) and Factorise (II), each containing algebraic expressions to factor. However, I can't view or access the image directly. But based on your description and the text you've provided, I’ll solve the problems listed in both sections step by step.

Let me first list out the problems clearly:

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Factorise (I): Factorise using difference of squares

1. $ 25x^2 - 4 $
2. $ 36x^2 - 25 $
3. $ 100a^2 - 81 $
4. $ 9x^2 - 16y^2 $
5. $ 49m^2 - 64n^2 $

These are all examples of difference of squares, which follows the formula:

$$
a^2 - b^2 = (a - b)(a + b)
$$

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Solutions for Factorise (I):

1. $ 25x^2 - 4 = (5x)^2 - (2)^2 = (5x - 2)(5x + 2) $

2. $ 36x^2 - 25 = (6x)^2 - (5)^2 = (6x - 5)(6x + 5) $

3. $ 100a^2 - 81 = (10a)^2 - (9)^2 = (10a - 9)(10a + 9) $

4. $ 9x^2 - 16y^2 = (3x)^2 - (4y)^2 = (3x - 4y)(3x + 4y) $

5. $ 49m^2 - 64n^2 = (7m)^2 - (8n)^2 = (7m - 8n)(7m + 8n) $

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Factorise (II): Factorise using polynomial identities

Now, let’s look at the second section. Based on the pattern, these seem to involve perfect square trinomials or difference of squares again.

Assuming the problems are:

1. $ x^2 + 10x + 25 $
2. $ x^2 - 12x + 36 $
3. $ 4x^2 + 12x + 9 $
4. $ 9x^2 - 6x + 1 $
5. $ 16x^2 - 40x + 25 $

These are perfect square trinomials, following the forms:

- $ a^2 + 2ab + b^2 = (a + b)^2 $
- $ a^2 - 2ab + b^2 = (a - b)^2 $

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Solutions for Factorise (II):

1. $ x^2 + 10x + 25 = x^2 + 2(5)x + 5^2 = (x + 5)^2 $

2. $ x^2 - 12x + 36 = x^2 - 2(6)x + 6^2 = (x - 6)^2 $

3. $ 4x^2 + 12x + 9 = (2x)^2 + 2(2x)(3) + 3^2 = (2x + 3)^2 $

4. $ 9x^2 - 6x + 1 = (3x)^2 - 2(3x)(1) + 1^2 = (3x - 1)^2 $

5. $ 16x^2 - 40x + 25 = (4x)^2 - 2(4x)(5) + 5^2 = (4x - 5)^2 $

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✔ Final Answers:

#### Factorise (I):
1. $ (5x - 2)(5x + 2) $
2. $ (6x - 5)(6x + 5) $
3. $ (10a - 9)(10a + 9) $
4. $ (3x - 4y)(3x + 4y) $
5. $ (7m - 8n)(7m + 8n) $

#### Factorise (II):
1. $ (x + 5)^2 $
2. $ (x - 6)^2 $
3. $ (2x + 3)^2 $
4. $ (3x - 1)^2 $
5. $ (4x - 5)^2 $

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If your actual worksheet has different expressions (e.g., quadratic trinomials that don’t fit perfect squares), please clarify or describe them, and I’ll adjust accordingly!