Question

Use a graphing calculator to approximate all of the functions real zeros. Round to 4 decimal places.
f(x)= 3x6−5x5−4x3+x2+x+1

Answer 1

@_alex_urena_
Have you graphed the function yet? How many zeroes do you see?

Answer 2

yeah i graphed it but what do you mean by zeros?

Answer 3

A zero in a polynomial (or any other) equation is where the graph intersects the x-axis. In general, a polynomial of 6th degree (your case) has a maximum of 6 zeroes.

Answer 4

yeah one of them is in between 0 and 1 and the other is on 2

Answer 5

Good, so there are two (real) zeroes.

Answer 6

Have you learned Newton's method, or the bisection method, or anything similar to those?

Answer 7

uhm i dont think so. never heard those names

Answer 8

Can you do trial and error?

Answer 9

whats that?

Answer 10

You're trying to solve for the zeroes of
f(x)=3x6−5x5−4x3+x2+x+1
which means that where f(x) crosses the x axis.
You know that there's one between 0 and 1.
so you can try with f(0)=1, f(1)=-3.
One is positive and the other is negative right?

Answer 11

yeah but idk where you got three from

Answer 12

f(1) means put x=1 in f(x), or
f(1)=3(1)^6−5(1)^5−4(1)^3+(1)^2+(1)+1=-3

Answer 13

ok i got that

Answer 14

That means the curve has to cross the x-axis (y=0) somewhere between 0 and 1.

Answer 15

ok

Answer 16

So you would try f(1/2), (between 0 and 1) to see if it is positive or negative.
Use your calculator to find f(1/2)=1.14.
So the solution f(x)=0 is between 1/2 and 1 because f(1/2)>0, and f(1)<0.

Answer 17

ok

Answer 18

So we have narrowed down the gap by a factor of two.
If you now repeat the same "trial" and "error", you can narrow down the zero to a much more accurate value.

Answer 19

brb

Answer 20

Your next try should be (0.5+1)/2=0.75.
So try f(0.75) and see what you have.
The key is to narrow down the gap.
If f(0.75) is positive, your next try is x=(0.75+1)/2=0.875, and if
f(0.75) is negative, your next try is x=(0.5+0.75)/2=0.625.