Question

How do you factor the expression `5x^2 - 16x + 3`?

Answer 1

`(5x-1)(x-3)`

Explanation:

Given: `5x^2-16x+3`

The 3 is a prime number so can only have {1, 3} as factors.
It is positive so the signs in the brackets are the same
The coefficient of x is negative (-16x) so they must both be negative.

The 5 is also a prime number so can only have {1, 5} as factors.

It is just a matter of getting them in the correct order

`color(blue)("Attempt 1")`

`(5x-3)(x-1) = 5x^2-8x ...color(red)("Fail")`

This will not work so there is no point in continuing with the rest of the multiplication.

`color(blue)("Attempt 2")`

`color(blue)((5x-1)(x-3))color(brown)(=5x^2-15x-x+3)color(blue)(=5x^2-16x+3)`