Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Second-degree, with zeros of −6 and 5, and goes to −∞ as x→−∞.

Answer 1

second degree means the highest exponent of the variable (x) is 2.

so, it is

y = ax² + bx + c

a zero simply means that y = 0.

so, the polynomial expression can be written as factors that turn to 0 at exactly the given values of x :

y = (x + 6)(x - 5)

when doing the multiplication :

y = x² - 5x + 6x - 30 = x² + x - 30

that would mean

a = 1

b = 1

c = -30

but something is not right yet, as the last piece of information means that y = -infinity, when x = -infinity (formally they both "go to infinity", because infinity can never be reached).

the x² term would normally bring everything to +infinity, as with growing and growing x, x² "drowns out" every impact of the x term and c, of course.

that means that "a" must be negative. e.g. -1

so, in our factoring we need to change one of the "x" to "-x" (so that we get "-x²") and then adapt the corresponding constant, so that the factor still turns to 0 at the given x-value :

y = (-x - 6)(x - 5)

after doing the multiplications

y = -x² + 5 - 6x + 30 = -x² - x + 30

a = -1

b = -1

c = 30

f(x) = y = -x² - x + 30