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 (1,12)

Answer 1

Step 1: Convert zeros to factors of the polynomial.

The allows us to convert zeros of -2, -1, and 3 into the factors (x+2)(x+1)(x-3).

This gives us the foundation of the polynomial, but it's missing a key piece, the leading coefficient which causes the vertical stretch/compression to hit the desired point.

So right now we have y = a · (x+2)(x+1)(x-3)

Step 2: Find "a".

This is where the point (1,12) comes in. We need to substitute x=1 and y=12 into the equation above to find a.

12 = a · (1+2)(1+1)(1-3)

12 = a · (3)(2)(-2)

12 = a · (-12)

-1 = a

That's the full function:

f(x) = -1 (x+2)(x+1)(x-3)