How do you show that #x^2+2x-3# and #2x^3-3x^2+7x-6# have a common linear factor?
Find their GCF, which turns out to be the linear polynomial
Let's consider the general case, since that gives the principle we will use:
Then we can long divide
#P_1(x) = Q_1(x)P_2(x) + R_1(x)#
So we can find the GCF of
Divide the polynomial of higher (or equal) degree by the one of lower degree to give a quotient and remainder.
If the remainder is
#0#then the divisor polynomial is the GCF.
Otherwise repeat with the remainder and the divisor polynomial.
So in our example:
#2x^3-3x^2+7x-6 = (x^2+2x-3)(2x-7)+27x-27#
#(2x^3-3x^2+7x-6)/(x^2+2x-3) = 2x-7" "#with remainder #27x-27#
#x^2+2x-3 = (x-1)(x+3)#
#(x^2+2x-3)/(x-1) = x+3" "#with no remainder
So the GCF is
Alternatively I could have factored both of the polynomials and simply identified the common factor.
The main reason I did not is that the method used above has the advantage of not requiring you to factor either of the polynomials.