Can someone show me how to do this? I got 5 more problems of the same kind. Thannk!

Consider the polynomial P(x), shown in both standard form and factored form.

P(x) = 1/2x^4 - 9/2x^3+21/2x^2+1/2x-15

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

State the behavior at the ends: (up/down) at the left, and (up/down) at the right?

State the y-intercept

State the x-intercept

Create a graph depicting the polynomial above.

setting each of the factors to zero gives you the x-intercepts

so the graph cuts the x-axis at -1,2,3, and 5

(a fourth degree polynomials can cut the x-axis in at most 4 places, a cubic in at most 3 places, a 5th degree in at most 5 places, etc)

If the highest term is an even power and positive the graph rises up in the first and second quadrants, if an even power but negative coefficient, it will drop down into the third and fourth quadrants.

If the highest power has an odd exponent, but the term is positive, the graph will rise in the first quadrant and drop into the third quadrant.

the y-intercept is when x=0, easy to see it must be -15

Fourth degree functions tend to look like a "W"

Can you sketch the graph ?

I was trying to can you prvide me with a picture so i can see where I have gone wrong? Thanks!

I am going to say that based on what you described the behavior of this graph is

up at the left, and down) at the right? Is this correct?

http://i263.photobucket.com/albums/ii157/mathmate/x4.png

I don't have a quick way to show the graph

I will describe how I would graph it.
As stated above, it will cross the x-axis at
-1,2,3, and 5
also (0,-15) is on the y-axis

using the factored version it is quite easy to see where the graph is at x=1 and x=4
I picked these values since they are roughly half-way between x-intercepts, and give you an idea how high or low the graph is between the x-intercepts.
I had (1,-8) and (4,-5)

so make your graph come down from the third quadrants, hit (-1,0), down to (0,-15) back up through (1,-8), then (2,0), rise up a bit and then down again to (3,0), downwards to roughly (4,-5), back up through (5,0) and then rising rapidly from there on.

I think i got it! Thanks!

To answer your question and solve your problem, here's how you can approach it step by step:

1. State the behavior at the ends:
To determine the behavior at the ends of the polynomial, you need to consider the leading term, which is the term with the highest exponent. In this case, the leading term is 1/2x^4.

- If the degree of the leading term is even (like in this case), and the coefficient of the leading term is positive (which it is), then the polynomial will have the same behavior at both ends, pointing upward (+) or "up-up."
- If the degree of the leading term is even (like in this case), and the coefficient of the leading term is negative, then the behavior at both ends will be pointing downward (-) or "down-down."
- If the degree of the leading term is odd, it will have different behaviors (+ at one end and - at the other end), but that is not the case here.

So, the behavior at the left (negative x) and the right (positive x) will both be "up" or (+).

2. State the y-intercept:
The y-intercept is a point on the graph where the polynomial intersects or crosses the y-axis. To find this, let x = 0 in the polynomial equation and solve for y:

P(0) = 1/2(0+1)(0-2)(0-3)(0-5)
= 1/2(1)(-2)(-3)(-5)
= 1/2(30)
= 15

So, the y-intercept is (0, 15).

3. State the x-intercepts:
The x-intercepts are the points on the graph where the polynomial intersects or crosses the x-axis. To find these, set P(x) equal to zero and solve for x.

Setting P(x) = 1/2(x+1)(x-2)(x-3)(x-5) = 0, we can solve for x by setting each factor equal to zero and solving:

x+1 = 0 => x = -1
x-2 = 0 => x = 2
x-3 = 0 => x = 3
x-5 = 0 => x = 5

So, the x-intercepts are (-1, 0), (2, 0), (3, 0), and (5, 0).

4. Create a graph depicting the polynomial:
To create the graph of the polynomial, plot the y-intercept and the x-intercepts on a coordinate plane. Since the polynomial has no other critical points provided, these points will give you an idea of the overall shape of the graph.

The graph will pass through the points: (-1, 0), (0, 15), (2, 0), (3, 0), and (5, 0). Based on the behavior at the ends (+), we can infer that the graph will go upward on the left and continue in an upward direction on the right.

Using this information, connect the points smoothly and draw a curve that follows the general shape of the graph. Make sure to label the axes, give the title "Graph of Polynomial P(x)," and indicate the x and y intercepts to create a proper graph.

Remember to use a significantly large enough scale on the x and y axes to ensure the graph is accurately depicted.

That's it! By following these steps, you can find the behavior at the ends, state the y-intercept and x-intercepts, and create a graph depicting the polynomial.