Difference between roots and factors of an equation?

Question

Relating to remainder and factor theorems

Caleb Alexander · Accepted Answer

See explanation below.

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.]

Sophie Adams · Answer

undefined The roots of an equation are the values of the variable that satisfy the equation, making it true. They are the solutions to the equation. On the other hand, the factors of an equation are the expressions that, when multiplied together, result in the equation. Factors can be used to express the equation in factored form. In summary, roots are the values that solve the equation, while factors are the expressions that, when multiplied, yield the equation.