Question

According to Descartes's rule of sign, how many possible positive and negative roots are there for the equation 0=4x7−2x4+2x3−4x−9 ?
Number of positive roots:
Number of negative roots:
0, 1 only, 2, only, 0 or 2. 3 only. 1 or 3

Answer 1

Answer:

$4x^7 -2x^4 + 2x^3 -4x-9$ has 1 or 3 Positive roots.

and 6 or 4 Negative roots.

Step-by-step explanation:

Descartes' rule of sign.

Function has the same number of positive real zeroes as there are number of changes in the sign of the coefficients, or an even number less than the found number.

Now, here the given expression is : $p(x) =4x^7 -2x^4 + 2x^3 -4x-9$

Here, the highest power of the variable x  = 7

So, the polynomial has TOTAL 7 ROOTS.

Now here, the number of sign changes are:

First : + to - in $+ 4x^7 \rightarrow -2x^4$

Second : - to + in $-2x^4 \rightarrow +2x^3$

Third: + to - in $+2x^3 \rightarrow -4x$

So, the total sign changes = 3

⇒ The total positive roots of p(x)  = 3 or a EVEN NUMBER LESS THAN 3  = 1

So, p(x) has either 3 or 1 POSITIVE ROOTS.

⇒ The number of negative roots p(x) has is either 4 or 6.

Hence  $p(x) =4x^7 -2x^4 + 2x^3 -4x-9$ has 1 or 3 Positive roots.

and 6 or 4 Negative roots.