How do you show that #x^2+2x3# and #2x^33x^2+7x6# have a common linear factor?
1 Answer
Find their GCF, which turns out to be the linear polynomial
Explanation:
Let's consider the general case, since that gives the principle we will use:
Suppose
Then we can long divide
#P_1(x) = Q_1(x)P_2(x) + R_1(x)#
Then since
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^33x^2+7x6 = (x^2+2x3)(2x7)+27x27#
That is:
#(2x^33x^2+7x6)/(x^2+2x3) = 2x7" "# with remainder#27x27#
Note that
#x^2+2x3 = (x1)(x+3)#
That is:
#(x^2+2x3)/(x1) = x+3" "# with no remainder
So the GCF is
Footnote
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.