Ask a question
0 0

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

How does the sign of the third term relate to the sign of the second term when
factoring? Please show me an example.

Tutors, please sign in to answer this question.

1 Answer

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.


Hassan H.