Question

How to factor `x^3-8x^2+15x`?

Answer 1

We can see that `x` is common to all the terms. We can write it as

`x*(x^2-8x+15)`

Now we need to factorise `color(blue)(x^2-8x+15`

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

It is in the form `ax^2 + bx + c` where `a=1, b=-8, c= 15`

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)*(15) and N_1+N_2 = -8`
`N_1*N_2 = 15 and N_1+N_2 = -8`

After Trial and Error, we get `N_1 = -3 and N_2 = -5`
`(-3)*(-5) = 15` and `(-3) + (-5) = -8`

So we can write the expression `x^2-8x+15` as
`color(blue) (x^2 - 3x - 5x+ 15)`
`=x(x-3)-5(x-3)`
`= (x-3)*(x-5)`

`color(green)(x^3-8x^2+15x = x*(x-3)*(x-5)`