Given the following algebraic expressions: 1.1 Fully factorise each expression 1.2 Determine the LCM of all four expressions. Simplify the following algebraic fractions: 2.1 2.2

$$a^2 - 6ab + 9b^2$$

This is a perfect square trinomial of the form $(x-y)^2 = x^2 - 2xy + y^2$. Here, $x=a$ and $y=3b$.

$$a^2 - 6ab + 9b^2 = \boxed{(a - 3b)^2}$$

$$a^2 - 9b^2$$

This is a difference of squares of the form $x^2 - y^2 = (x-y)(x+y)$. Here, $x=a$ and $y=3b$.

$$a^2 - 9b^2 = \boxed{(a - 3b)(a + 3b)}$$

$$a^2 - 3ab$$

Factor out the common term $a$.

$$a^2 - 3ab = \boxed{a(a - 3b)}$$

$$3a^2 - 6ab - 9b^2$$

First, factor out the common numerical factor, which is 3.

$$3a^2 - 6ab - 9b^2 = 3(a^2 - 2ab - 3b^2)$$

Now, factor the quadratic expression $a^2 - 2ab - 3b^2$. We look for two terms that multiply to $-3b^2$ and add to $-2b$. These are $-3b$ and $b$.

$$a^2 - 2ab - 3b^2 = (a - 3b)(a + b)$$

So, the fully factorized expression is:

$$3a^2 - 6ab - 9b^2 = \boxed{3(a - 3b)(a + b)}$$

2.1.2 Determine the LCM of all four expressions.

The factorized expressions are:

1. $(a - 3b)^2$
2. $(a - 3b)(a + 3b)$
3. $a(a - 3b)$
4. $3(a - 3b)(a + b)$

The unique factors are:

• Numerical factor: 3 (from expression 4)
• Algebraic factors: $a$, $(a - 3b)$, $(a + 3b)$, $(a + b)$

The highest powers for each unique factor are:

• $3^1$
• $a^1$
• $(a - 3b)^2$ (from expression 1)
• $(a + 3b)^1$ (from expression 2)
• $(a + b)^1$ (from expression 4)

$$\text{LCM} = 3 \cdot a \cdot (a - 3b)^2 \cdot (a + 3b) \cdot (a + b)$$ $$\text{LCM} = \boxed{3a(a - 3b)^2(a + 3b)(a + b)}$$