*****10 POINTS!!!!***** Help with precalculus regarding zeros and x-intercepts!?

Question

Hi, I'm stuck on the following problem (and have been for like an hour):

R(x) = .5x^3 - 2x^2 + 3
~~~~ -----------------------
~~~~~ .1x^3 + x - 3

(NOTE: The little ~ things are just to keep everything lined up.)

I'm stuck on how you find the x-intercepts. I know it's using the numerator and graphing it, etc., but how do you do it on a TI-83 Plus calculator?

Also, I'm supposed to find the "real zeros" which I know involves the denominator, and I know the answer is something like 2, I know what a zero of a function is -- I just don't know how you get the .1x^3 + x - 3 into a format where you can factor it into the zeros.

Please help!!! I'll give 10 points for best answer! Thank you so much!

Answer 1

To find the x-intercepts, set the numerator equal to zero.

If you're using the TI-83 plus calculator, do the following.

1. Type in the equation

(.5x^3-2x^2+3)/(.1x^3+x-3)

in the Y= screen

2. Graph it, and Zoom to best fit

3. From the Calc menu, select Zeros or Roots.

4. You have to do this for each zero... go slightly to the left of the zero, and hit enter for the left bound. Then go slight to the right of the zero, and hit enter for the right bound. If it asks for a guess, just hit enter.

5. It'll spit it out at the bottom of the screen. Repeat for any other zeros.

To find the number of real zeros for this function, count the number of zeros you come up with (I counted 3).

If you use just the numerator to find the zeros, you may come up with extraneous zeros (i.e. - zeros that aren't real). You would use the numerator if you didn't have a calculator. Since you are allowed to use one, plot the actual rational expression, instead of just the numerator.


Below is all the zeros for the function so you can check your answer.

(-1.08613, 0)
(1.57199, 0)
(3.51414, 0)

Good luck!

Answer 2

Firstly, the zeros of the function and the x intercepts are the same thing. In the case of a rational function they are found from the numerator so just put 0.5x^3 - 2x^2 + 3 = 0 and solve it. I would immediately multiply by 2 to get rid of fractions, x^3 - 4x^2 + 6 = 0. Solving a cubic equation involves a certain amount of guesswork. Oops, I was thinking in pre-calculator days. Nowadays you have it so easy.

The denominator being zero gives you points on the x axis where the function is undefined. Each side of this the function will head off to infinity positive or negative. If any value of x found in numerator is the same as one found in the denominator, then this is a factor that could have been cancelled out, but this is unlikely,

Answer 3

OK, so you've figured out that the task is to find out when .5x^3 - 2x^2 + 3 = 0.

I don't have a TI83+, I have a TI84+ - but I bet the process for doing this is virtually (if not entirely) identical. Here's how you do it on a TI84+:
- press the "y=" button
- Enter the equation .5x^3 - 2x^2 + 3
- press the "graph" button
- press the "calc" button. (That's 2nd-trace)
- select "zero"
- use the arrow keys to put the left bound on the left side of one of your x-intercepts, then press enter.
- use the arrow keys to put the right bound on the right side of that x-intercept, then press enter.
- press enter again, to "guess".
The calculator will then give you the coordinates of the x-intercept.

Repeat the last few steps with each x-intercept - placing bounds on both sides of them, one at a time, and then calculating.

Oh yeah... to be technically precise, make sure the denominator isn't zero at any of your x-intercepts. Can't have you dividing by zero, after all.

Hope that helps!

Answer 4

pretend. -2 is the y intercept. that's healthier to split both variables on both area of the ='s signal first: 4x - 5y - 10 = 0 4x -4x - 5y - 10 = 0 - 4x 0 -5y -10 + 10 = -4x + 10 -5y + 0 = -4x + 10 5y = 4x - 10 y = 4/5x - 2 The x intercept is the point on the line the position it crosses the x axis. At this aspect, y=0. replace 0 for y to discover the x intercept: 0 = 4/5x - 2 2 = 4/5x - 2 + 2 x = (5/4)*2 = 10/4 = 5/2 = 2.5 The y intercept is the point on the line which crosses the y axis. Set x = 0 to discover the y intercept: y = 4/5x - 2 y = 0 - 2 y = -2