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How does the sign of the third term in the trinomial ax²+bx+c affects the factor.

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Hello Harry,

When factoring a quadratic f(x) = ax2+bx+c, you are (usually) seeking integers p and q which satisfy

p+q = b

pq   = ac

simultaneously.  You can check to see whether factoring over the integers is worthwhile by seeing whether the discriminant (b2-4ac) is a perfect square.  If so, then try to factor.

Now, if the signs of the coefficients a and c are the same, then the two numbers p and q must both be positive if b is positive, and both be negative if b is negative (why?), whereas if the signs of a and c differ, then p and q must differ in sign.  You may be able to find more information if you study certain examples carefully.

Here is a simple one:

f(x) = 2x2-3x-2.

First, the discriminant is in this case (-3)2-4(2)(-2) = 25, which is a square, so factoring over the integers can be pursued.  In this case, a = 2 and c = -2 differ in sign, so the integers p and q sought must differ in sign.  If we complete the factorization (by whatever method you are used to), we find that p = -4 and q = 1, so that

f(x) = 2x2-3x-2 = 2x2+px+qx-2 = 2x2-4x+x-2 = (2x+1)(x-2).

Observe that the roots of f(x) (-1/2 and 2) also differ in sign.  You may want to see if you can state this as a general rule, based on what we just discussed.

I hope this puts you on the path to the solution you seek.

Regards,

Hassan H.