Question

Difference between roots and factors of an equation?

Relating to remainder and factor theorems

Answer 1

See explanation below.

Explanation:

The 'difference between roots and factors of an equation' is not a straightforward question. Let's define both to establish the link between the two..

Assume we have some function of a single variable `x`;
we'll call this `f(x)`

Then we can form an equation: `f(x) =0`

Then the "roots" of this equation are all the values of `x` that satisfy that equation. Remember that these values may be real and/or imaginary.

Now, up to this point we have not assumed anything about `fx)`. To consider factors, we now need to assume that `f(x) = g(x)*h(x)`.

That is that `f(x)` factorises into some functions `g(x) xx h(x)`

If we recall our equation: `f(x)=0`
Then we can now say that either `g(x) =0 or h(x)=0`

.. and thus show the link between the roots and factors of an equation.

[NB: A simple example of these general principles would be where `f(x)` is a quadratic function that factorises into two linear factors.]