Question

how you would analyze the zeros of the polynomial function f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 using Descartes’ Rule of Signs

Answer 1

I don't understand what it's asking for is it asking to find positive negitive and complex?

Answer 2

@sara1234 It's asking for no. of positive or negative zeros.!!

Answer 3

huh? for no of?

Answer 4

Yeah, Using Descartes' Rule of Signs we can determine that. I'll explain you

Answer 5

We have the polynomial
\[f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19\]
First step it to count the no. of sign changes in f(x)
I'll underline wherever there is a sign change between two consecutive terms
i.e from + to - or - to +
\[f(x) = –3x^5 \underline{– 8x^4 +25x^3} – 8x^2 +x – 19 \longrightarrow 1st\ change\]
\[f(x) = –3x^5 – 8x^4 \underline{+25x^3 – 8x^2} +x – 19 \longrightarrow 2nd\ change\]
\[f(x) = –3x^5 – 8x^4 +25x^3 \underline{– 8x^2 +x} – 19 \longrightarrow 3rd\ change\]
\[f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 \underline{+x – 19} \longrightarrow 4th\ change\]
so there are 4 sign changes, do you get this part?

Answer 6

isnt there 5 sign changes?

Answer 7

oh never miind i got it

Answer 8

So there could be 4 positive zero, However, some of the roots may be generated by the Quadratic Formula, and these pairs of roots may be complex.
so we have to down count by 2.
Either it has 4 or 2 or 0 positive roots.
Do you get this?

Answer 9

yes

Answer 10

Now can you write f(-x), it's used to find no. of negative zeros

Answer 11

-3(-x)^5-8(-x)^4+25^3-8(-x)^2+(-x)-19

Answer 12

Good , we need to simplify it. I'll write that
\[f(-x)=-3(-x)^5-8(-x)^4+25(-x)^3-8(-x)^2+(-x)-19\]
\[f(-x)=3x^5-8x^4-25x^3-8x^2-x-19\]
do you get this?

Answer 13

Yes i do

Answer 14

Now count the no. of sign changes here

Answer 15

only one

Answer 16

Yeah so it's having 1 negative zero, we can't down count by 2, it'll become -1.
So it has 1 negative zero and 4 or 2 or 0 positive zeros.
Do you get this?

Answer 17

Yeah thankssss!! Now we find complex?

Answer 18

We found that there are either 4 or 2 or 0 positive roots.
But these could also be complex, so
if we have have 4 positive roots there will be 0 complex
if we have 2 positive there will be 2 complex
or if we have 0 positive there will be 4 complex

Answer 19

why do we have to add the ) positive?

Answer 20

0

Answer 21

Because 0 could be due to 4 complex roots

Answer 22

theres 4 or 2 positive real zeros but why do we have to add the 0

Answer 23

There could be a double quadratic, with 4 roots complex. therefore we won't have any positive zero

Answer 24

thanks for all your help @ash2326 you really helped me :)

Answer 25

you're welcome. did you understand all the things?

Answer 26

Yessss :D