If x^2+bx+c is factorable and c is negative, then the signs of the last term factors of the binomials are opposite, why?
I x^2+bc+c is factorable and c is positive, then the signs of the last term factors of the binomials are the same, Why?
Am not understanding this. could someone please explain to me.
To understand why the signs of the last term factors of the binomials differ in the first scenario and are the same in the second scenario, we need to consider the concept of factoring quadratic expressions.
When factoring a quadratic expression of the form x^2 + bx + c, we are essentially trying to find two binomial factors that, when multiplied together, give us the original quadratic expression. These binomial factors have the form (x + p) and (x + q), where p and q are constants.
For the first scenario, where c is negative, let's assume that p and q have opposite signs, where p is positive and q is negative.
When we expand (x + p)(x + q), we get x^2 + (p + q)x + pq.
Since pq is negative and c is negative, this means that (p + q)x should be positive in order to get the correct sum of bx.
Therefore, we need p to be greater in magnitude than q to satisfy these conditions. This is why the signs of the last term factors of the binomials are opposite.
For the second scenario, where c is positive, let's assume that p and q have the same sign, where both p and q are positive.
When we expand (x + p)(x + q), we get x^2 + (p + q)x + pq.
Since pq is positive and c is positive, this means that (p + q)x should be positive in order to get the correct sum of bx.
Therefore, we need p and q to be positive to satisfy these conditions. This is why the signs of the last term factors of the binomials are the same.
In conclusion, the signs of the last term factors of the binomials differ or are the same based on whether the constant term (c) in the quadratic expression is negative or positive, respectively.