Factorising 2x³-x² -13x-6

To factorise the given expression, we need to find two binomials that multiply together to give the original expression.

Step 1: Grouping

We start by grouping the terms in pairs. We can group the first two terms and the last two terms as follows:

2x³ - x² - 13x - 6 = (2x³ - x²) + (-13x - 6)

Step 2: Factoring out the common factors

Now, we factor out the common factors from each pair of terms.

From the first pair, we can factor out x²:
2x³ - x² = x²(2x - 1)

From the second pair, we can factor out -1:
-13x - 6 = -1(13x + 6)

Step 3: Factoring by grouping

Now, we can factor by grouping. We have x²(2x - 1) + (-1)(13x + 6).

We can factor out a common factor of (2x - 1) from the first two terms, and a common factor of (13x + 6) from the last two terms:

x²(2x - 1) + (-1)(13x + 6) = (2x - 1)(x² - 13x - 6)

Step 4: Factoring the quadratic expression

The quadratic expression x² - 13x - 6 can be factored further. We need to find two numbers that multiply to give -6 and add up to -13. After some trial and error, we find that -15 and 2 satisfy these conditions:

x² - 13x - 6 = (x - 15)(x + 2)

Final Answer:

Putting it all together, the factorisation of 2x³ - x² - 13x - 6 is:

2x³ - x² - 13x - 6 = (2x - 1)(x - 15)(x + 2)

Summary:

To factorise the expression 2x³ - x² - 13x - 6, we followed the steps of grouping, factoring out common factors, factoring by grouping, and factoring the quadratic expression. By applying these steps, we obtained the factorised form of (2x - 1)(x - 15)(x + 2).