Question

How do you factorise `2x^3-5x+x+2`?

Answer 1

I find it best to look at the polynomial is the following form:
`2x^3+0x^2-4x+2`

Explanation:

The best way is by long division. However, finding a factor of this polynomial is difficult.

...

After some trial and error, you will realise that `x-1` is a factor.
Therefore mathematically:
`2x^3+0x^2-4x+2`
`=(2x^3-2x^2)+(2x^2-2x)-(2x+2)`
`=(x-1)(2x^2+2x-2)`
`=2(x-1)(x^2+x-1)`

I hope this helps!

Answer 2

Demonstrating long division

Explanation:

Given that it is determined that `x=1` is a solution (by observation)

Giving `(x-1)("something")`

Using place keepers of 0 value. Example `0x^2`

Then we have:

`color(white)("ddddddddd.dd")2x^3+0x^2-4x+2`
`color(magenta)(2x^2)(x-1)-> ul(2x^3-2x^2color(white)("ddd")larr" Subtract")`
`color(white)("ddddddddddddd")0+2x^2-4x+2`
`color(magenta)(2x)(x-1) -> color(white)("dddd.d")ul(2x^2-2x larr" Subtrtact")`
`color(white)("dddddddddddddddddd")0-2x+2`
`color(magenta)(-2)(x-1)-> color(white)("ddddddd.d")ul(-2x+2)`

Putting it all together:

`(x-1)(color(magenta)(2x^2+2x-2))`

`2(x-1)(x^2+x-2)`

If you wished to take the quadratic 'down' further you would use the formula.