the following trinomial given to us:
Find all the integer values for b so that the trinomial can be factored how do I do this can you show your work please!
This particular trinomial is a quadratic. There are lots of ways to factor a quadratic: note that factoring it into two linear factors is the same as finding the two roots of the equation formed by setting it equal to zero.
One way is as follows. Suppose 3x^2 + bx + 2 = (px+q)(rx+s) with p,q,r,s integers. Then pr = 3 and qs = 2. That is, p=3,r=1 or p=1,r=3 or p=-3,r=-1 or p=-1,r=-3; similarly q=2,s=1 or q=1,s=2 or q=-2,s=-1 or q=-1,s=-2. The value of b is ps+qr which is therefore one of 7,5,-7,-5.
Another way is as follows, Solve the quadratic to give x = (-b +- sqrt(b^2-24) ) / 12. We need b^2-24 to be a perfect square, say d^2. If b^2-24 = d^2 then b^2-d^2 =24, so (b+d)(b-d)=24. The possibilities for b+d,b-d respectively are 24,1 or 12,2 or 8,3 or 6,4 or 4,6 or 3,8 or 2,12 or 1,24 or -24,-1 or ... -1,-24. Again we find that the integral values of b are 7,5,-5,-7.
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