How do you show that #x^2+2x-3# and #2x^3-3x^2+7x-6# have a common linear factor?

1 Answer
Jan 28, 2017

Answer:

Find their GCF, which turns out to be the linear polynomial #x-1#

Explanation:

Let's consider the general case, since that gives the principle we will use:

Suppose #P_1(x)# and #P_2(x)# are polynomials with a common polynomial factor #P(x)#.

Then we can long divide #P_1(x)# by #P_2(x)# to find a quotient polynomial #Q_1(x)# and remainder polynomial #R_1(x)# with degree less than #P_2(x)#:

#P_1(x) = Q_1(x)P_2(x) + R_1(x)#

Then since #P_1(x)# and #P_2(x)# are both multiples of #P(x)#, #R_1(x)# must also be a multiple of #P(x)# and has lower degree than #P_2(x)#. Note that scalar factors are not important to us in this context. If #P(x)# is a factor then any non-zero scalar multiple of it is too (and vice versa).

So we can find the GCF of #P_1(x)# and #P_2(x)# by the following method:

  • 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.

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So in our example:

#2x^3-3x^2+7x-6 = (x^2+2x-3)(2x-7)+27x-27#

That is:

#(2x^3-3x^2+7x-6)/(x^2+2x-3) = 2x-7" "# with remainder #27x-27#

Note that #27x-27 = 27(x-1)#, so for tidiness, let's divide by #27# before proceeding.

#x^2+2x-3 = (x-1)(x+3)#

That is:

#(x^2+2x-3)/(x-1) = x+3" "# with no remainder

So the GCF is #(x-1)#.

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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.