Expanding and Factorising Revision | PDF - Free Printable

Problem: Solving the Expanding and Factorising Revision Tasks

The provided document contains several tasks related to expanding expressions, simplifying them, and factorising. Below, I will solve each part step by step.

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Section 1: Expand

#### a) \( 5(x + 2) \)
- Distribute \( 5 \) across the terms inside the parentheses:
\[
5(x + 2) = 5 \cdot x + 5 \cdot 2 = 5x + 10
\]

#### b) \( 5(3x - 4) \)
- Distribute \( 5 \) across the terms inside the parentheses:
\[
5(3x - 4) = 5 \cdot 3x + 5 \cdot (-4) = 15x - 20
\]

#### c) \( 2x(x + 4) \)
- Distribute \( 2x \) across the terms inside the parentheses:
\[
2x(x + 4) = 2x \cdot x + 2x \cdot 4 = 2x^2 + 8x
\]

#### d) \( 7x^2(3x - 4y) \)
- Distribute \( 7x^2 \) across the terms inside the parentheses:
\[
7x^2(3x - 4y) = 7x^2 \cdot 3x + 7x^2 \cdot (-4y) = 21x^3 - 28x^2y
\]

#### e) \( -2|x - 3| \)
- Note: The absolute value \( |x - 3| \) is not expanded further unless specific values for \( x \) are given. For now, we leave it as:
\[
-2|x - 3|
\]

#### f) \( -x|x + y| \)
- Similarly, the absolute value \( |x + y| \) is not expanded further unless specific values for \( x \) and \( y \) are given. For now, we leave it as:
\[
-x|x + y|
\]

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Section 2: Expand and Simplify

#### a) \( 4(x + 3) + 6(x + 2) \)
- First, expand each term:
\[
4(x + 3) = 4x + 12, \quad 6(x + 2) = 6x + 12
\]
- Combine the results:
\[
4(x + 3) + 6(x + 2) = (4x + 12) + (6x + 12) = 4x + 6x + 12 + 12 = 10x + 24
\]

#### b) \( 5(x - 2) + 3(x - 1) \)
- First, expand each term:
\[
5(x - 2) = 5x - 10, \quad 3(x - 1) = 3x - 3
\]
- Combine the results:
\[
5(x - 2) + 3(x - 1) = (5x - 10) + (3x - 3) = 5x + 3x - 10 - 3 = 8x - 13
\]

#### c) \( 3(2x + 3) - 4(x + 1) \)
- First, expand each term:
\[
3(2x + 3) = 6x + 9, \quad 4(x + 1) = 4x + 4
\]
- Combine the results:
\[
3(2x + 3) - 4(x + 1) = (6x + 9) - (4x + 4) = 6x + 9 - 4x - 4 = 2x + 5
\]

#### d) \( 5(2x + 5) - 6(x - 2) \)
- First, expand each term:
\[
5(2x + 5) = 10x + 25, \quad 6(x - 2) = 6x - 12
\]
- Combine the results:
\[
5(2x + 5) - 6(x - 2) = (10x + 25) - (6x - 12) = 10x + 25 - 6x + 12 = 4x + 37
\]

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Section 3: Factorise

#### a) \( 6x + 24 \)
- Factor out the greatest common factor (GCF), which is \( 6 \):
\[
6x + 24 = 6(x + 4)
\]

#### b) \( 10x + 25 \)
- Factor out the GCF, which is \( 5 \):
\[
10x + 25 = 5(2x + 5)
\]

#### c) \( 12y + 16y^2 \)
- Factor out the GCF, which is \( 4y \):
\[
12y + 16y^2 = 4y(3 + 4y)
\]

#### d) \( 6x^2y + 21xy^2 \)
- Factor out the GCF, which is \( 3xy \):
\[
6x^2y + 21xy^2 = 3xy(2x + 7y)
\]

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Section 4: Expand and Simplify

#### a) \( (x + 3)(x + 2) \)
- Use the distributive property (FOIL method):
\[
(x + 3)(x + 2) = x(x + 2) + 3(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
\]

#### b) \( (x + 6)(x + 9) \)
- Use the distributive property (FOIL method):
\[
(x + 6)(x + 9) = x(x + 9) + 6(x + 9) = x^2 + 9x + 6x + 54 = x^2 + 15x + 54
\]

#### c) \( (x - 2)(x + 4) \)
- Use the distributive property (FOIL method):
\[
(x - 2)(x + 4) = x(x + 4) - 2(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8
\]

#### d) \( (x - 4)(x - 5) \)
- Use the distributive property (FOIL method):
\[
(x - 4)(x - 5) = x(x - 5) - 4(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20
\]

#### e) \( (x + 5)(x - 5) \)
- Use the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[
(x + 5)(x - 5) = x^2 - 5^2 = x^2 - 25
\]

#### f) \( (2x + 1)(x - 4) \)
- Use the distributive property (FOIL method):
\[
(2x + 1)(x - 4) = 2x(x - 4) + 1(x - 4) = 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4
\]

#### g) \( (2x + 5)(2x - 5) \)
- Use the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[
(2x + 5)(2x - 5) = (2x)^2 - 5^2 = 4x^2 - 25
\]

#### h) \( (3x + 4)(4x + 2) \)
- Use the distributive property (FOIL method):
\[
(3x + 4)(4x + 2) = 3x(4x + 2) + 4(4x + 2) = 12x^2 + 6x + 16x + 8 = 12x^2 + 22x + 8
\]

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Section 5: Factorise

#### a) \( x^2 + 6x + 8 \)
- Find two numbers that multiply to \( 8 \) and add to \( 6 \). These numbers are \( 4 \) and \( 2 \):
\[
x^2 + 6x + 8 = (x + 4)(x + 2)
\]

#### b) \( x^2 + 5x + 6 \)
- Find two numbers that multiply to \( 6 \) and add to \( 5 \). These numbers are \( 3 \) and \( 2 \):
\[
x^2 + 5x + 6 = (x + 3)(x + 2)
\]

#### c) \( x^2 + 10x + 16 \)
- Find two numbers that multiply to \( 16 \) and add to \( 10 \). These numbers are \( 8 \) and \( 2 \):
\[
x^2 + 10x + 16 = (x + 8)(x + 2)
\]

#### d) \( x^2 - 3x + 2 \)
- Find two numbers that multiply to \( 2 \) and add to \( -3 \). These numbers are \( -2 \) and \( -1 \):
\[
x^2 - 3x + 2 = (x - 2)(x - 1)
\]

#### e) \( x^2 - 8x + 7 \)
- Find two numbers that multiply to \( 7 \) and add to \( -8 \). These numbers are \( -7 \) and \( -1 \):
\[
x^2 - 8x + 7 = (x - 7)(x - 1)
\]

#### f) \( x^2 + 2x - 8 \)
- Find two numbers that multiply to \( -8 \) and add to \( 2 \). These numbers are \( 4 \) and \( -2 \):
\[
x^2 + 2x - 8 = (x + 4)(x - 2)
\]

#### g) \( x^2 - 9 \)
- Use the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \):
\[
x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)
\]

#### h) \( x^2 - 3x - 28 \)
- Find two numbers that multiply to \( -28 \) and add to \( -3 \). These numbers are \( -7 \) and \( 4 \):
\[
x^2 - 3x - 28 = (x - 7)(x + 4)
\]

#### i) \( x^2 - x - 30 \)
- Find two numbers that multiply to \( -30 \) and add to \( -1 \). These numbers are \( -6 \) and \( 5 \):
\[
x^2 - x - 30 = (x - 6)(x + 5)
\]

#### j) \( x^2 - 100 \)
- Use the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \):
\[
x^2 - 100 = x^2 - 10^2 = (x + 10)(x - 10)
\]

#### k) \( x^2 + 5x - 14 \)
- Find two numbers that multiply to \( -14 \) and add to \( 5 \). These numbers are \( 7 \) and \( -2 \):
\[
x^2 + 5x - 14 = (x + 7)(x - 2)
\]

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Final Answers:

1. Expand:
- a) \( 5x + 10 \)
- b) \( 15x - 20 \)
- c) \( 2x^2 + 8x \)
- d) \( 21x^3 - 28x^2y \)
- e) \( -2|x - 3| \)
- f) \( -x|x + y| \)

2. Expand and Simplify:
- a) \( 10x + 24 \)
- b) \( 8x - 13 \)
- c) \( 2x + 5 \)
- d) \( 4x + 37 \)

3. Factorise:
- a) \( 6(x + 4) \)
- b) \( 5(2x + 5) \)
- c) \( 4y(3 + 4y) \)
- d) \( 3xy(2x + 7y) \)

4. Expand and Simplify:
- a) \( x^2 + 5x + 6 \)
- b) \( x^2 + 15x + 54 \)
- c) \( x^2 + 2x - 8 \)
- d) \( x^2 - 9x + 20 \)
- e) \( x^2 - 25 \)
- f) \( 2x^2 - 7x - 4 \)
- g) \( 4x^2 - 25 \)
- h) \( 12x^2 + 22x + 8 \)

5. Factorise:
- a) \( (x + 4)(x + 2) \)
- b) \( (x + 3)(x + 2) \)
- c) \( (x + 8)(x + 2) \)
- d) \( (x - 2)(x - 1) \)
- e) \( (x - 7)(x - 1) \)
- f) \( (x + 4)(x - 2) \)
- g) \( (x + 3)(x - 3) \)
- h) \( (x - 7)(x + 4) \)
- i) \( (x - 6)(x + 5) \)
- j) \( (x + 10)(x - 10) \)
- k) \( (x + 7)(x - 2) \)

\boxed{\text{See detailed solutions above.}}