# How do you use Descartes rule of signs to analyze all possibilities for the complex zeroes of: #y= x^6 - 3x^5 - 5x^4 + 11x^3 - 48x^2 + 92x - 48#?

##### 1 Answer

There are 5 or 3 or 1 positive real roots

There is 1 negative real root

There are 2 imaginary roots

#### Explanation:

From the given polynomial equation,

We can see that the number of sign difference

+ to - count 1

- to + count 1

+ to - count 1

- to + count 1

+ to - count 1

So there are 5 sign differences. These means there are 5 or 3 or 1 positive real roots. The number of sign differences is equal to the number of positive real roots or less than that by an even number. So if its not 5 then it may 3 or 1.

For the negative roots

We have to substititute -x in place of x then inspect the changes in sign.

use -x

The result is

Now let us check the sign difference

+ to - count 1

There is 1 negative real root or less than that by an even number.

The rest are imaginary roots.

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Actually, the roots of the polynomial

are

God bless....I hope the explanation is useful.