To determine the signs of the factors for the trinomial [tex]\( x^2 - bx - c \)[/tex], where both signs in the trinomial are negative, let's analyze the situation step-by-step.
1. Understanding the Components of the Trinomial:
- The given trinomial is [tex]\( x^2 - bx - c \)[/tex].
- This is a quadratic expression with a leading coefficient of 1 (as seen in [tex]\( x^2 \)[/tex]), a middle term [tex]\(-bx\)[/tex], and a constant term [tex]\(-c\)[/tex].
2. Factor Form:
- For a trinomial [tex]\( x^2 - bx - c \)[/tex], it can be factored into the form [tex]\( (x - p)(x + q) \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are constants.
3. Determining the Signs:
- The product of the constants [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must equal the constant term in the trinomial, which is [tex]\(-c\)[/tex].
- Similarly, the sum [tex]\( p + q \)[/tex] must equal the coefficient of the middle term, which is [tex]\(-b\)[/tex].
4. Analyzing the Impact of Signs:
- Since the constant term is [tex]\(-c\)[/tex], it indicates that [tex]\( p \cdot q \)[/tex] must be negative. This implies that one of the factors must be positive and the other must be negative. Therefore, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have opposite signs.
5. Conclusion:
- Given that the product of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is negative ([tex]\(-c\)[/tex]), the possible signs for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be such that one is positive and the other is negative.
Therefore, the signs of the factors of the trinomial [tex]\( x^2 - bx - c \)[/tex] are:
C. one positive and one negative
