Factorise:
`3z ^(2) - 10z + 8`

Answer

To factorise the expression \(3z^2 - 10z + 8\), follow these steps:

### Step 1: Identify the coefficients
The expression is in the form \(az^2 + bz + c\), where:
- \(a = 3\) (coefficient of \(z^2\))
- \(b = -10\) (coefficient of \(z\))
- \(c = 8\) (constant term)

### Step 2: Multiply \(a\) and \(c\)
Multiply the coefficient of \(z^2\) (which is \(3\)) by the constant term (which is \(8\)):
\[ 3 \times 8 = 24 \]

### Step 3: Find two numbers that multiply to \(24\) and add to \(-10\)
We need to find two numbers that multiply to \(24\) and add to \(-10\). The two numbers are:
\[ -6 \text{ and } -4 \]
because:
\[ -6 \times -4 = 24 \quad \text{and} \quad -6 + (-4) = -10 \]

### Step 4: Rewrite the middle term
Rewrite the expression \(3z^2 - 10z + 8\) using the two numbers found:
\[ 3z^2 - 6z - 4z + 8 \]

### Step 5: Group the terms
Group the terms in pairs:
\[ (3z^2 - 6z) + (-4z + 8) \]

### Step 6: Factor out the common factors in each group
Factor out the common factors from each group:
\[ 3z(z - 2) - 4(z - 2) \]

### Step 7: Factor out the common binomial
Now, notice that \((z - 2)\) is common in both terms:
\[ (3z - 4)(z - 2) \]

### Final Answer
Thus, the factorised form of \(3z^2 - 10z + 8\) is:
\[ (3z - 4)(z - 2) \]

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