Question

How do you simplify `(x - 2) ( x - 5) ( x - 7) = 8\cdot 5\cdot 3`?

Answer 1

Assuming that we are restricted to Real values:
`color(white)("XXX")color(red)(x=10)`

Explanation:

#{: ((x-2)(x-5)(x-7)," = ",8 * 5 * 3), (x^3-14x^2+59x-70," = ",120) :}#

`rarr x^3-14x^2+59x-190=0`

Using the Rational Factor Theorem:
any roots to this equation must be `in {+-1,+-2,+-5,+-10,+-19,+-38,+-95,+-190}`

Setting up a spreadsheet to perform Synthetic Division for these values (I've only shown the first few):

We find that `x=10` is a root and
`x^3-14x^2+59x-190=(x-10)(1x^2-4x+19)`

Checking the discriminant of `(1x^2-4x+19)` we see that there are no further Real roots. (Use the quadratic formula if you want to find the other 2 Complex roots).

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We have that
`color(white)("xxx")(x-2)(x-5)(x-7)=8 * 5 * 3`
is equivalent to
`color(white)("xxx")(x-10)(x^2-4x+19)=0`

and since `(x^2-4x+19)!=0` for any Real value of `x`
we can divide both sides by `(x^2-4x+19)` to get
`color(white)("xxx")x-10=0`
or
`color(white)("xxx")x=10`