Question

How do you simplify `(4g^2 - 64g + 252)/(g-7)`?

Answer 1

Let's Factorise the `color(red)(NUMERATOR)` first.

The Numerator is `color(red)(4g^2 - 64g + 252)`

`= 4(g^2 - 16g + 63)` (4 was the factor common to all terms)

Now we need to factorise `color(blue) (g^2 - 16g + 63)`

We can use Splitting the Middle Term technique to factorise this.

It is in the form `ax^2 + bx + c` where `a=1, b=-16, c= 63`

To split the middle term, we need to think of two numbers `N_1 and N_2` such that:
`N_1*N_2 = a*c and N_1+N_2 = b`
`N_1*N_2 = (1)*(63) and N_1+N_2 = 16`
`N_1*N_2 = 63 and N_1+N_2 = -16`

After Trial and Error, we get `N_1 = -7 and N_2 = -9`
`(-7)*(-9) = 63` and `(-7) + (-9) = -16`

So we can write the expression in blue as
`color(blue) (g^2 - 7g -9g+ 63)`
`=g(g-7)-9(g-7)`
`= (g-7)*(g-9)`

The Numerator can be written as `color(red)(4(g-7)*(g-9))`

The expression we have been given is
`(4g^2 - 64g + 252)/(g-7)`

After the numerator was factorised, the Expression can now be written as :

`(4(g-7)*(g-9))/(g-7)`

`=(4*cancel((g-7))*(g-9))/cancel((g-7))`

`= 4*(g-9)`

`(4g^2 - 64g + 252)/(g-7)` = `4*(g-9)`