Am not understanding this. Could someone please explain to me.
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?
If 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?
Proofread your questions before you post them. I assume in the second instance, you mean x^2+bx+c.
Positive times negative = negative
Positive times positive or negative times negative = positive.
To understand why the signs of the last term factors of the binomials are opposite when c is negative in the expression x^2+bx+c, and why they are the same when c is positive in the expression x^2+bc+c, we need to consider the factoring process for these quadratic expressions.
When we factor a quadratic expression, we try to find two binomial factors that, when multiplied together, will give us the original expression. In the case of x^2+bx+c, we need to find two binomial factors in the form (x+a)(x+b) that multiply to give us x^2+bx+c.
Let's first consider the case when c is negative. This means that when we multiply the binomial factors (x+a)(x+b), we need their product to have a negative constant term. In other words, a*b must be negative.
To achieve this, we need one factor to be positive and the other factor to be negative. If both factors were positive, their product would be positive. Similarly, if both factors were negative, their product would also be positive. Therefore, to make the product negative, we need one factor to be positive and the other factor to be negative. This is why the signs of the last term factors of the binomials are opposite when c is negative.
Now let's consider the case when c is positive. This means that when we multiply the binomial factors (x+a)(x+b), we need their product to have a positive constant term. In other words, a*b must be positive.
To achieve this, we need both factors to have the same sign. If one factor is positive and the other is negative, their product would be negative. Therefore, to make the product positive, we need both factors to be either positive or negative. This is why the signs of the last term factors of the binomials are the same when c is positive.
In summary, the signs of the last term factors of the binomials are opposite when c is negative to achieve a negative product, and they are the same when c is positive to achieve a positive product. This is a result of the factoring process and the need to find binomial factors that produce the desired constant term.