Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Third-degree, with zeros of −2, −1, and 3, and passes through the point (4,6

Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Third-degree, with zeros of −2, −1, and 3, and passes through the point (4,6

Answer 1

Let's start with the zeros part.

If a polynomial p(x) has a zero at some particular value like 3, it means that the binomial (x-3) is one of its factors. Meaning, we can write it as p(x)=(x-3)g(x), where g(x) is some further polynomial expression. Since we're actually given three zeros here, we can write p(x)=(x+2)(x+1)(x-3)g(x), where again, g(x) is some further polynomial expression we have yet to find. You can check that x=-2, x=-1, and x=3 all cause this expression to be 0.

Now we use the fact that we're looking for a third degree polynomial. This tells us that, in fact, g(x) is at most some constant (that I'll now relabel a), because (x+2)(x+1)(x-3) is ALREADY a third degree polynomial. If g(x) contains any factors of x, it will raise us to a fourth order polynomial or higher. So, p(x)=a(x+2)(x+1)(x-3), where a is some constant.

To find what a is, we use the last given information, that the curve must pass through the point (4,6). Substituting 4 in for x, we get

p(4)=a(4+2)(4+1)(4-3)=a(6)(5)(1)=30a=6

So, a=6/30, or 1/5

In total, we have

p(x)=(1/5)(x+2)(x+1)(x-3)

You may expand this out if you prefer.

p(x)=(1/5) x³ - (7/5) x - 6/5