Question

How do you simplify `(6t^3 + 5t^2 + 9) / (2t + 3)`?

Answer 1

`3t^2 - 2t^2 + 3`

Explanation:

`(6t^3 + 5t^2 + 9) / (2t+3)`

`(2t+3(3t^2-2t+3)) / (2t+3)`

• 2t+3 on numerator and denomoniator cancel out.

ANSWER: `3t^2 - 2t^2 + 3`

You have to factorise the top that `2t + 3` directly factorises `6t^3 + 5t^2 + 9` so that it cancels the denominator giving you a value of `3t^2 - 2t^2 + 3`.

The are many ways of factorising `6t^3 + 5t^2 + 9`, polynomial division, rational root theorem.

I would recommend polynomial division as its pretty straight forward to learn and rely on. Here is a link that will teach you all about it and you can practise with: http://www.purplemath.com/modules/polydiv2.htm

Answer 2

`3t^2-2t+3`

Explanation:

`"one way is to use the divisor as a factor in the numerator"`

`"consider the numerator"`

`color(red)(3t^2)(2t+3)color(magenta)(-9t^2)+5t^2+9`

`=color(red)(3t^2)(2t+3)color(red)(-2t)(2t+3)color(magenta)(+6t)+9`

`=color(red)(3t^2)(2t+3)color(red)(-2t)(2t+3)color(red)(+3)(2t+3)cancel(color(magenta)(-9))cancel(+9)`

`=color(red)(3t^2)(2t+3)color(red)(-2t)(2t+3)color(red)(+3)(2t+3)+0`

`rArr(cancel((2t+3))(3t^2-2t+3))/cancel((2t+3))=3t^2-2t+3`