Question

Use Descartes rule to determine the possible number of positive and negative solutions for the polynomial: f(x)=10x^4-21x^2+11 If more than one solution is possible type the smaller solution first.The number of positive solutions is Answer for part 1 and coordinate 1 or Answer for part 1 and coordinate 2The number of negative solutions is Answer for part 2 and coordinate 1 or Answer for part 2 and coordinate 2

Answer 1

Descartes’s rule of signs: The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by an even number.

Hence the coefficients are 10,−21, 11.

Since there 2 signs changes, then we have

`2\text{ or 0 positive root }`

Therefore,

There 2 or 0 positive root

To find the number of negative real roots, substitute x with −x in the given polynomial:

Hence the equation becomes

`\begin{gathered} x=-x \\ 10(-x)^4-21(-x)^2+11 \\ =10x^4-21x+11 \end{gathered}`

The coefficients are 10,−21, 11, Then there are 2 signs changes

Hence we have

`2\text{ or no negative root real root }`

There are 2 or 0 negative root

Answer; There 2 or 0 positive root