Math CentralQuandaries & Queries


Question from Mukulu, a student:

What is the value of a if 2x2-x-6 ,3x2-8x+4 and ax3-10x-4 have a common factor.

Hi Mukulu,

If all the polynomials have common factor, we can find the common factor(s) of the first two and use the remainder theorem to find the value of "a". If I look at 2x2-x-6 it can broken into factors of

2x2-x-6 = 2x2-4x+3x-6 = 2x(x-2)+3(x-2)=(2x+3)(x-2)

Since 2x2-x-6 has factors of (2x+3)& (x-2) we know it has roots at x=-3/2 & x=2 and the other two polynomials must at least one of these roots. So let's use the remainder theorem to find if x=-3/2 or x=2 are roots of 3x2-8x+4

f(-3/2)=3(-3/2)2-8(-3/2)+4= 22.75

f(2)=3(2)2-8(2)+4= 0

Since x=2 yields a remainder of 0 for 3x2-8x+4 we know that (x-2) is the only common factor for all of the polynomials. If you substitute x=2 into your last equation and set it equal to 0, you can solve for a using simple algebra.

Hope this helps,


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