Question

Factorise the following
$\large\underline{\bf{ \red{ {2x}^{3} - {4x}^{2} + 3x - 6 - 2xy + y}}}$
​

Answer 1

To Facotrise :

• $\large\underline{\bf{ \red{ {2x}^{3} - {4x}^{2} + 3x - 6 - 2xy + y}}}$

Solution

$\sf \: 2x³-4x²+ 3x-6-2xy +y$

$\sf = (2x³-4x²)+(3x-6)-(2xy-y)$

$\sf = 2x²(x-2)+3(x-2)-2y(x-2)$

$\sf=(x-2)(2x²-2y+3)$

Answer 2

We can write in terms of $\rm y$.

$\rm2x^{3}-4x^{2}+3x-6-2xy+y$

$\;$

$\rm=(-2x+1)y+(2x^{3}-4x^{2}+3x-6)$

$\;$

We observe that the constant term is factorizable.

$\rm 2x^{3}-4x^{2}+3x-6$

$\;$

$\rm=2x^{2}(x-2)+3(x-2)$

$\;$

$\rm=(2x^{2}+3)(x-2)$

$\;$

Given polynomial is,

$\rm(-2x+1)y+(2x^{3}-4x^{2}+3x-6)$

$\;$

$\rm=(-2x+1)y+(2x^{2}+3)(x-2)$

$\;$

There is no common factor. We cannot write the polynomial in terms of the product of factors, which situation is called irreducible.

$\;$

Hence,

$\boxed{\rm2x^{3}-4x^{2}+3x-6-2xy+y\ is\ irreducible.}$