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# Find an Equation of the Polynomial Function - Problem 2 3,137 views

###### Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.