Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.

Second-degree, with zeros of −4 and 6, and goes to −∞ as x→−∞.

p(x) =

Answer 1

Answer: p(x) = -x² + 2x + 24

Proof:

We can solve this by first setting up the formula p(x) = (ax+b)(cx+d)

Because the function goes to goes to −∞ as x→−∞, we know a or c must be negative. There are many values that could work, but we will use the easiest ones which are a = -1 and b = 1

Now we have p(x) = (-x+b)(x+d)

Next, find b and d values that make p(-4) = 0 and p(6) = 0.

If either (-x+b) is equal to 0 or (x+d) is equal to zero, the the output will be 0. (-(-4)-4) = 0 and (6-6) = 0, so we know b = -4 and d = -6

Now we have p(x) = (-x-4)(x-6)

Optionally convert this to y = ax² + bx + c form:

p(x) = -x² + 2x + 24