Factorisation worksheet with questions and video tutorial link.

Problem: Factorisation of Expressions

The task is to factorise the given algebraic expressions. Factorisation involves rewriting an expression as a product of its factors. Here's how we can solve each part step by step.

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#### Question 1: Factorise the following expressions

We will factorise each expression by identifying the greatest common factor (GCF) of the terms.

##### (a) \( 4x + 6 \)
- The GCF of \( 4x \) and \( 6 \) is \( 2 \).
- Factor out \( 2 \):
\[
4x + 6 = 2(2x + 3)
\]

##### (b) \( 15x + 20 \)
- The GCF of \( 15x \) and \( 20 \) is \( 5 \).
- Factor out \( 5 \):
\[
15x + 20 = 5(3x + 4)
\]

##### (c) \( 9y - 12 \)
- The GCF of \( 9y \) and \( 12 \) is \( 3 \).
- Factor out \( 3 \):
\[
9y - 12 = 3(3y - 4)
\]

##### (d) \( 5x + 15 \)
- The GCF of \( 5x \) and \( 15 \) is \( 5 \).
- Factor out \( 5 \):
\[
5x + 15 = 5(x + 3)
\]

##### (e) \( 6x - 3 \)
- The GCF of \( 6x \) and \( 3 \) is \( 3 \).
- Factor out \( 3 \):
\[
6x - 3 = 3(2x - 1)
\]

##### (f) \( 4x + 8 \)
- The GCF of \( 4x \) and \( 8 \) is \( 4 \).
- Factor out \( 4 \):
\[
4x + 8 = 4(x + 2)
\]

##### (g) \( 5y - 25 \)
- The GCF of \( 5y \) and \( 25 \) is \( 5 \).
- Factor out \( 5 \):
\[
5y - 25 = 5(y - 5)
\]

##### (h) \( 8w + 24 \)
- The GCF of \( 8w \) and \( 24 \) is \( 8 \).
- Factor out \( 8 \):
\[
8w + 24 = 8(w + 3)
\]

##### (i) \( 10y + 15 \)
- The GCF of \( 10y \) and \( 15 \) is \( 5 \).
- Factor out \( 5 \):
\[
10y + 15 = 5(2y + 3)
\]

##### (j) \( 14w + 21 \)
- The GCF of \( 14w \) and \( 21 \) is \( 7 \).
- Factor out \( 7 \):
\[
14w + 21 = 7(2w + 3)
\]

##### (k) \( 20y - 30 \)
- The GCF of \( 20y \) and \( 30 \) is \( 10 \).
- Factor out \( 10 \):
\[
20y - 30 = 10(2y - 3)
\]

##### (l) \( 27x + 18 \)
- The GCF of \( 27x \) and \( 18 \) is \( 9 \).
- Factor out \( 9 \):
\[
27x + 18 = 9(3x + 2)
\]

##### (m) \( 6 - 4x \)
- The GCF of \( 6 \) and \( 4x \) is \( 2 \).
- Factor out \( 2 \):
\[
6 - 4x = 2(3 - 2x)
\]

##### (n) \( 9 + 12y \)
- The GCF of \( 9 \) and \( 12y \) is \( 3 \).
- Factor out \( 3 \):
\[
9 + 12y = 3(3 + 4y)
\]

##### (o) \( 45 + 60x \)
- The GCF of \( 45 \) and \( 60x \) is \( 15 \).
- Factor out \( 15 \):
\[
45 + 60x = 15(3 + 4x)
\]

##### (p) \( 16y - 32 \)
- The GCF of \( 16y \) and \( 32 \) is \( 16 \).
- Factor out \( 16 \):
\[
16y - 32 = 16(y - 2)
\]

##### (q) \( 22a + 55 \)
- The GCF of \( 22a \) and \( 55 \) is \( 11 \).
- Factor out \( 11 \):
\[
22a + 55 = 11(2a + 5)
\]

##### (r) \( 100 - 40y \)
- The GCF of \( 100 \) and \( 40y \) is \( 20 \).
- Factor out \( 20 \):
\[
100 - 40y = 20(5 - 2y)
\]

##### (s) \( 6x + 9y \)
- The GCF of \( 6x \) and \( 9y \) is \( 3 \).
- Factor out \( 3 \):
\[
6x + 9y = 3(2x + 3y)
\]

##### (t) \( 4w - 2a \)
- The GCF of \( 4w \) and \( 2a \) is \( 2 \).
- Factor out \( 2 \):
\[
4w - 2a = 2(2w - a)
\]

##### (u) \( 25y - 35z \)
- The GCF of \( 25y \) and \( 35z \) is \( 5 \).
- Factor out \( 5 \):
\[
25y - 35z = 5(5y - 7z)
\]

##### (v) \( 8x^2 + 20 \)
- The GCF of \( 8x^2 \) and \( 20 \) is \( 4 \).
- Factor out \( 4 \):
\[
8x^2 + 20 = 4(2x^2 + 5)
\]

##### (w) \( 30y^3 - 15 \)
- The GCF of \( 30y^3 \) and \( 15 \) is \( 15 \).
- Factor out \( 15 \):
\[
30y^3 - 15 = 15(2y^3 - 1)
\]

##### (x) \( 42y + 28x - 56c \)
- The GCF of \( 42y \), \( 28x \), and \( 56c \) is \( 14 \).
- Factor out \( 14 \):
\[
42y + 28x - 56c = 14(3y + 2x - 4c)
\]

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#### Question 2: Factorise the following expressions

These expressions involve variables raised to powers. We will factor out the common variable and numerical factors.

##### (a) \( x^2 + 7x \)
- The GCF of \( x^2 \) and \( 7x \) is \( x \).
- Factor out \( x \):
\[
x^2 + 7x = x(x + 7)
\]

##### (b) \( x^2 - 3x \)
- The GCF of \( x^2 \) and \( 3x \) is \( x \).
- Factor out \( x \):
\[
x^2 - 3x = x(x - 3)
\]

##### (c) \( y^2 + y \)
- The GCF of \( y^2 \) and \( y \) is \( y \).
- Factor out \( y \):
\[
y^2 + y = y(y + 1)
\]

##### (d) \( w^2 + 9w \)
- The GCF of \( w^2 \) and \( 9w \) is \( w \).
- Factor out \( w \):
\[
w^2 + 9w = w(w + 9)
\]

##### (e) \( x^2 - 7x \)
- The GCF of \( x^2 \) and \( 7x \) is \( x \).
- Factor out \( x \):
\[
x^2 - 7x = x(x - 7)
\]

##### (f) \( 4w^2 + 10w \)
- The GCF of \( 4w^2 \) and \( 10w \) is \( 2w \).
- Factor out \( 2w \):
\[
4w^2 + 10w = 2w(2w + 5)
\]

##### (g) \( 6x^2 - 8x \)
- The GCF of \( 6x^2 \) and \( 8x \) is \( 2x \).
- Factor out \( 2x \):
\[
6x^2 - 8x = 2x(3x - 4)
\]

##### (h) \( 9y^2 - 6y \)
- The GCF of \( 9y^2 \) and \( 6y \) is \( 3y \).
- Factor out \( 3y \):
\[
9y^2 - 6y = 3y(3y - 2)
\]

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Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{(a)} & 2(2x + 3) \\
\text{(b)} & 5(3x + 4) \\
\text{(c)} & 3(3y - 4) \\
\text{(d)} & 5(x + 3) \\
\text{(e)} & 3(2x - 1) \\
\text{(f)} & 4(x + 2) \\
\text{(g)} & 5(y - 5) \\
\text{(h)} & 8(w + 3) \\
\text{(i)} & 5(2y + 3) \\
\text{(j)} & 7(2w + 3) \\
\text{(k)} & 10(2y - 3) \\
\text{(l)} & 9(3x + 2) \\
\text{(m)} & 2(3 - 2x) \\
\text{(n)} & 3(3 + 4y) \\
\text{(o)} & 15(3 + 4x) \\
\text{(p)} & 16(y - 2) \\
\text{(q)} & 11(2a + 5) \\
\text{(r)} & 20(5 - 2y) \\
\text{(s)} & 3(2x + 3y) \\
\text{(t)} & 2(2w - a) \\
\text{(u)} & 5(5y - 7z) \\
\text{(v)} & 4(2x^2 + 5) \\
\text{(w)} & 15(2y^3 - 1) \\
\text{(x)} & 14(3y + 2x - 4c) \\
\text{(a)} & x(x + 7) \\
\text{(b)} & x(x - 3) \\
\text{(c)} & y(y + 1) \\
\text{(d)} & w(w + 9) \\
\text{(e)} & x(x - 7) \\
\text{(f)} & 2w(2w + 5) \\
\text{(g)} & 2x(3x - 4) \\
\text{(h)} & 3y(3y - 2) \\
\end{array}
}
\]