Question

for the given function, determine consecutive values of x between which each real zero is located f(x)= -19x^4-3x^3-12x^2+20x+25 just explain how to get the problem :)

Answer 1

The idea here is that on one side of the zero, f(a) will be positive, and on the other, f(b) will be negative. As this function is continuous, that means there must be a value of x somewhere between a and b where f(x) = 0.

Pick a couple of integer values of x between -3 and 3 and evaluate that function.You should be able to figure out where those zeros must be, approximately.

Answer 2

The problem is that I don't understand what a zero means? could you explain

Answer 3

A zero is a spot where the function equals...wait for it....drum roll please...0 :-)

Answer 4

zeros are also called roots.

Answer 5

lol so once I find zeros in the functions what do I do?

Answer 6

Well, finding the zeros is not necessarily all that easy. Could you solve \[-19x^4-3x^3-12x^2+20x+25 = 0\]

Answer 7

Umm no but I can plug the different numbers in to see if it equals zero if that's what you mean by solving

Answer 8

Often there isn't an exact solution, and it has to be found numerically. One of the methods used is essentially picking a pair of points, evaluating the function at each of them, deciding if it must have a zero between (because one side is positive and one is negative), and sashaying up and down the x-axis looking for such spots. Now, if you actually try evaluating this function, you'll discover that as you leave the vicinity of -2 < x < 2 the value of the function gets really, really big, so the roots are going to be in that range.

Answer 9

In case you think I'm kidding about it getting really big, check out the graph I've attached.

Answer 10

so how would I put that in answer form The answer choices are
a. there are zeros between x=1 and x=2, x-0 and x=1
b. there are zeros between x=2 and x=3, x=1 and x=0, x=-1 and x=-2, x=-2 and x=-3
c. x=0 and x=1
d. there is a zero between x=0 and x=-1

Answer 11

Oh, that makes it much easier. You've got a bunch of values of x to plug in. If the sign of the function changes between x=1 and x=2, that means there's a zero between x = 1 and x = 2. Make yourself a table showing the sign of the value of the function at each of the values of x listed in the answer possibilities, and then work out where the zeros must be. It's not that hard, you just have to evaluate the polynomial successfully.

Answer 12

oh ok so I just plug the values in too see if there is a sign change consecutively in the polynomial

Answer 13

Looking at a closeup of the graph, I think you maybe didn't copy the answers quite correctly, but that doesn't matter so long as you look at the right ones when you make your choice.

Answer 14

Do you know Descartes' Rule of Signs? That will allow you to figure out how many real roots there are for this equation.

Answer 15

Sure do and I only see one sign change in the original polynomial

Answer 16

so that means 1 real positive root. how about negative roots (sign changes in f(-x))

Answer 17

easy way to construct f(-x) is just to flip the signs on the odd power roots.

Answer 18

ok I only see one on that to

Answer 19

okay, so 1 real positive, 1 real negative, how many roots does this polynomial have in total?

Answer 20

2

Answer 21

nope. what's the highest power of x here?

Answer 22

oh sorry lol 4

Answer 23

right, so 1 real positive, 1 real negative, 2 complex (in a conjugate pair)

Answer 24

oh so once I have 1 neg and 1 pos I just put the other two in conjugate pairs

Answer 25

that probably eliminates some answer choices right there. anyhow, you should go through the exercise of figuring out the value of polynomial at each of those points. if you want, I can check your values for you

Answer 26

well, the other two don't matter for this problem — they asked only for real roots/zeros

Answer 27

oh ok

Answer 28

I have another similar problem if you want me to do it

Answer 29

finish this one first, I have to cook dinner, and then I'll be back.

Answer 30

ok thanx for the help

Answer 31

if you add 0 to f(x) then the outcome will become: 25
if you add 1 to f(x0 then the outcome will become: 11
if you add 2 to f(x) then the outcome is -311
if you add 3 to f(x) you get -1954

Answer 32

good, now do the negative values as well...

Answer 33

ok

Answer 34

for -1 it will be -23
for -2 it will be -343
for -3 it will be -1601

Answer 35

very good. so here they are in a convenient table:

\[\begin{array}{cc}
\text{x} & \text{y}\\-3 & -1601 \\
-2 & -343 \\
-1 & -23 \\
0 & 25 \\
1 & 11 \\
2 & -311 \\
3 & -1643 \\
\end{array}\]

Between which consecutive values of \(x\) does \(y\) change sign?

Answer 36

is it 2 times

Answer 37

Yes, but I need to know where it changes.

x=-3 y negative
x=-2 y negative
x=-1 y negative
x=0 y positive HELLO

what does that imply about the path of the curve between x = -1 and x = 0?

Answer 38

oh ok its saying that that's a real zero right because the sign changes. And that's why there are 4 numbers

Answer 39

Okay, so there's a zero between x = -1 and x = 0. Not sure exactly where, but we could repeat this process with the values of x spaced closer together, right?

Now continuing on through the table, where is the next place where the sign of the y values changes?

Answer 40

between x=1 and x=2

Answer 41

OMG I get it now

Answer 42

yep, I think you do :-)

Answer 43

Here's a zoomed in version of the graph...

Answer 44

YAY I'm happy now but do you think you could help me with one more question

Answer 45

The solutions of the equation are pretty interesting :-)

Answer 46

that looks really confusing but I'm only in algebra 2 sooooo....

Answer 47

No, your instincts are good :-)

Answer 48

the numeric versions are: {{x -> -0.268551 - 1.2515 I}, {x -> -0.268551 +
1.2515 I}, {x -> -0.7264}, {x -> 1.10561}}

Answer 49

ok then so do you think you could help me with a whole new problem or are you tired

Answer 50

1 positive real root at about 1.10561, 1 negative real root at -0.7264, and the two complex roots off in never-never land.

Answer 51

Yes, why don't you post as a fresh problem and tag me, I'll be back in about 3 minutes

Answer 52

ok

Answer 53

estimate the x-coordinates at which the relative maxima and relative minima occur for the function.
f(x)= 7x^3-2x^2+3
The answers are
a. the relative maximum is at x=0, and the relative minimum is at x=0.19
b. the relative maximum is at x=1, and the relative minimum is at x=0.19
c. the relative maximum is at x=0, and the relative minimum is at x=-0.19
d. the relative maximum is at x=1, and the relative minimum is at x=-0.19

Answer 54

okay, here I think you just evaluate the function at x = 0, x = 1, x = 0.19, x = -0.19 and draw your conclusions

Answer 55

so im going to plug it in the function

Answer 56

yeah.

Answer 57

but if I plug in 0 I will get 3 has an answer

Answer 58

how will that help me find the min or max

Answer 59

Okay. You're going to find the values at x = -0.19, x = 0, x = 0.19, x = 1
and the compare the y values. the highest one will be a relative maximum, right? and the lowest one will be a relative minimum.

Answer 60

it's too simple, you're overthinking it :-)

Answer 61

ok so since I got 3 that's going to be a value on 0

Answer 62

yes, that's the value at x=0.

\[\begin{array}{l|l|l}
& \text{x} & \text{y} \\
\hline
& -0.19 & 2.87979 \\

& 0 & 3 \\
& 0.19 & 2.97581 \\
& 1 & 8 \\
\end{array}\]

Answer 63

well from here it seems like the answers are going too be d. 1 and -0.19 :) I fully comprehend now

Answer 64

vertical lines are at \(x=\pm 0.19\)

Answer 65

that's an odd degree polynomial and it only has 1 real zero

Answer 66

Yep. Complex conjugate roots at \(0.439638\pm0.542339i\)

Answer 67

uhmmmm and I know im going to make an A on my test tomorrow thanx for your help

Answer 68

Good luck with the exam — let me know how you do!

Answer 69

ok and thanx again gn