Consider the following function. f(x) = x³ - 3x²

Consider the following function. f(x) = x³ - 3x² - 16x + 48 (a) Find all real zeros of the polynomial function. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) (b) Determine whether the multiplicity of each zero is even or odd. smaller x-value larger x-value (c) Determine the maximum possible number of turning points of the graph of the function. Use a graphing utility to graph the function and verify your answers. turning point(s)

Transcript

00:01 This problem gives us the function f of x equals 3x to the 4th plus 9x squared plus 6 and we want to answer a few questions about it 1, 0s and multiplicity of those zeros.

00:09 Also what the max turning points would be for this function and then we want to confirm all that with the graph.

00:14 So for zeros and multiplicity to find the zeros i'm going to try to factor this expression when it's set equal to 0.

00:20 So we have 3x to the 4 plus 9x squared plus 6 equal to 0.

00:26 So we can factor out a 3 which leaves us with x to the 4th plus 3x squared plus 2 and then to factor this further we can kind of forget about the 3 if we're just finding zeros because it doesn't have an x value with it and when we factor this we can use the same rules that we normally do with the trinomial where we put an x and an x in a front and say what two numbers multiply be the last value in this case 2 and add to be the middle value 3 because this is leading coefficients now 1 the only difference is since this is x to the 4th you need to write x squared in front instead of just x but then you just still ask yourself the same question it's what two numbers, multiple to be two and had to be three.

01:02 And that's a positive two and a positive one.

01:06 And that's as far as you can factor this, because x squared plus two can be factored, and an x squared plus one is not a difference.

01:12 If it was a difference of squares, then we could factor it further, but that's a plus.

01:15 And when we set these equal to zero, to try to find our zeros or x intercepts, the same thing happens for both.

01:24 We get an x squared or x squared equal to negative numbers and square roots of negative numbers are not real zeros, they're imaginary.

01:33 So for the zeros, there are none.

01:38 And if there are no zeros, we don't need to worry about the multiplicity, because the multiplicity would help us to determine the behavior at those zeros...

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