# Factorising this damned equation

• StephenP91
In summary, Stephen is seeking help with factoring the polynomial 2x^3 - 6x^2 + 2 = 0 in Pure Core 1 Mathematics without using a calculator. He has tried using the remainder theorem and the rational root theorem with no success. He is looking for a simple factor that is not a non-integer and wants to plot the graph by finding where it crosses the x-axis. He suggests possibly factoring 2x^3 - 6x^2 into 2x^2(x-3) and then subtracting 2 from the y-coordinates to find the coordinates of the roots.
StephenP91
Well, it's only Pure Core 1 Mathematics. I am trying to factorise:

2x^3 - 6x^2 + 2 = 0

Now, you can't use a calculator. I've tried finding a factor using the remainder theorem, but I just can't find a simple one (|x<5|). I am sure they don't expect us to use a complicated number, like a non-integar. So I just need someone to help me with factorising this.

Thank you,
Stephen.

Remainder theorem?

There is the rational root theorem which gives you a short list of things to try -- and if your polynomial has any rational roots, they must appear on this list.

That said, it's "easy" to see that this polynomial doesn't have any linear factors. Can you give an argument that it doesn't have any quadratic factors?

However, I guarantee that this polynomial has a nontrivial factor... so if it's not linear, and it's not quadratic, what must it be?

P.S. by "factorize", I assume you mean to factor over the integers -- i.e. you want each factor to be a polynomial with integer coefficients.

By factorise I mean, place intro brackets so that I may find the information I am looking for. Namely where the graph crosses the X axis so that I can plot the graph.

I was thinking though. Could I just factorise 2x^3 - 6x^2 into 2x^2(x-3) and then get the points I need, then from that subtract 2 to each of the y co-ords to get the co-ords of each of the roots?

## What is factorising?

Factorising is a mathematical process in which an equation is broken down into its simplest form by identifying common factors or terms.

## Why do we need to factorise equations?

Factorising equations can help us solve them more easily, as it simplifies the equation and makes it easier to work with. It also allows us to find the roots or solutions of the equation.

## How do we factorise an equation?

To factorise an equation, we need to identify common factors or terms and then use mathematical techniques such as grouping, difference of squares, or trial and error to simplify the equation.

## What are the benefits of factorising an equation?

Factorising an equation can help us solve it more efficiently, as it reduces the amount of calculations needed. It also allows us to find the solutions or roots of the equation, which can be useful in real-world applications.

## What are some common mistakes to avoid when factorising an equation?

Some common mistakes to avoid when factorising an equation include not factoring out the greatest common factor, forgetting to include negative signs, and incorrectly grouping terms. It is also important to check that the factors are correct by expanding them back to the original equation.

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