Question

In general, what must be true about the value of c if zero pairs are needed to model a polynomial of the form x^2 + bx + c?

A) c < 0
B) c > 0
C) c = 0
D) c can be any real number

Answer 1

For a polynomial of the form x^2 + bx + c, zero pairs are needed to model it if c is equal to 0. This is because the polynomial then has a root at zero, as indicated by the factored form x(x + b) with the multiplicative property of zero.

Step-by-step explanation:

In general, for a polynomial of the form x^2 + bx + c, zero pairs are needed to model it if the constant c must be equal to 0. This is because when c is equal to zero, the polynomial can be factored as x(x + b), indicating that one of the factors (and therefore one of the roots) is 0. This aligns with the multiplicative property of zero which states that any number multiplied by zero equals zero. Therefore, if zero pairs are needed, it implies that the polynomial will be touched or cross the x-axis at the origin, making the correct answer C) c = 0.

When we have values for c that are different from zero, this results in either two real roots (if c > 0 and b^2 - 4ac > 0), one real root (if b^2 - 4ac = 0), or two complex roots (if c < 0 and b^2 - 4ac < 0).