It may determine that there were no negative real roots through using Descartes' rule of the sign. Only one option with no negative noughts shown is the one mentioned below, and further calculation can be defined as follows:
Given:
[tex]\bold{f(x) = x^3 - 14x^2 + 61x - 84}\\\\\bold{Factor=} (x-7)\\\\[/tex]
To find:
find zeros of function=?
Solution:
when factor=0
then
[tex]\to \bold{x-7=0}\\\\\to \bold{x=7}\\\\[/tex]
putting the value of x into the given function:
[tex]\to \bold{f(x) = x^3 - 14x^2 + 61x - 84}\\\\[/tex]
[tex]\to \bold{f(7) = 7^3 - 14\times 7^2 + 61\times 7 - 84}\\\\\to \bold{f(7) = 343 - 14\times 49 + 427 - 84}\\\\\to \bold{f(7) = 343 - 686 + 427 - 84}\\\\\to \bold{f(7) = 770- 770}\\\\\to \bold{f(7) =0}\\\\[/tex]
and When x=3:
[tex]\to \bold{f(3) = 3^3 - 14\times 3^2 + 61\times 3 - 84}\\\\\to \bold{f(3) = 27 - 14\times 9 + 183 - 84}\\\\\to \bold{f(3) = 27 - 126 + 183 - 84}\\\\\to \bold{f(3) = 210- 210}\\\\\to \bold{f(3) =0}\\\\[/tex]
and When x=4:
[tex]\to \bold{f(4) = 4^3 - 14\times 4^2 + 61\times 4 - 84}\\\\\to \bold{f(4) = 64- 14\times 16 + 244- 84}\\\\\to \bold{f(4) = 64 - 224 + 244- 84}\\\\\to \bold{f(3) = 308- 308}\\\\\to \bold{f(3) =0}\\\\[/tex]
Therefore the final choice "7,4,3".
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Please find garph in the attached file.
