Question

When simplifying 4x^3-10x^2+6x over 2x^3+x^2-3x, what are the term(s) that can be cancelled

Answer 1

Cancelling x and (2x-3), we get, the simplest form as

`(4x^3-10x^2+6x)/(2x^3+x^2-3x)=(2(x-1))/((x+1))`

Explanation:

Consider the given two expressions
`4x^3-10x^2+6x` and
`2x^3+x^2-3x`.

We solve both expressions seperately,

Consider the first expression
`4x^3-10x^2+6x`

Taking x common from the expression,

`4x^3-10x^2+6x=x(4x^2-10x+6)`

The terms in brackets is a quadratic equation, we can solve using middle term splitting method,

`4x^2-10x+6`

-10x can be written as -4x-6x , we get,

`4x^2-10x+6=4x^2-4x-6x+6`

`\Rightarrow 4x(x-1)-6(x-1)=(4x-6)(x-1)`

`4x^3-10x+6)=2x(2x-3)(x-1)`

Consider the second term ,
`2x^3+x^2-3x`

Taking x common from the expression we have,

`2x^3+x^2-3x=x(2x^2+x-3)`

The terms in brackets is a quadratic equation, we can solve using middle term splitting method,

`2x^2+x-3`

x can be written as 3x-2x

`2x^2+x-3=2x^2+2x-3x-3`

`2x^2+2x-3x-3=2x(x+1)-3(x+1)=(2x-3)(x+1)`

Thus,
`2x^3+x^2-3x=x(2x-3)(x+1)`

Our expression is
`4x^3-10x^2+6x` over
`2x^3+x^2-3x`

is
`(4x^3-10x^2+6x)/(2x^3+x^2-3x)`

`(4x^3-10x^2+6x)/(2x^3+x^2-3x)=(2x(2x-3)(x-1))/(x(2x-3)(x+1))`

Cancelling same terms from numerator and denominator , thus cancelling x and (2x-3), we get, the simplest form as

`(4x^3-10x^2+6x)/(2x^3+x^2-3x)=(2(x-1))/((x+1))`