Find an Equation of the Polynomial Function - Problem 2

Let’s find the equation for this quintic polynomial, q of x which looks like this. Now quintic means degree 5. Remember the number of turning points, the maximum number of that a polynomial can have is the degree minus 1, in this case 4 and it has exactly 4; 1, 2, 3, 4. So it’s a really wavy graph.

Let’s start by identifying the intercepts. The intercepts are -3, -1 and 2 and that gives us factors of x plus 3, x plus 1 and x minus 2. Now notice at this intercept and at this intercept, we’ve got that quadratic behavior and that indicates that we’re going to have a square here and at 2 also we’re going to have a square. But now at -1, just passes right through.

The other thing we have to take care of is the stretch factor. We have to determine what the stretch a factor is, see if the graph will pass through (0, 2). Let’s plug in (0, 2). When we plug in zero we get a times zero plus 3, 3 squared, zero plus 1 is 1, zero minus 2, -2² and that has to equal 2. So we’ve got 4 times 9, 36a equals 2. A has to be 1/18. That’s pretty much it, you could now just write your polynomial function, q of x is 1/18. We have x plus 3 squared, x plus 1 and x minus 2 squared.