x^4+16x^2+7x-11=0 Descartes’ rule of signs to find the nature of the roots of the equation
max 1 positive root: f(x) has one sign change
max 1 negative root: f(-x) has one sign change
To apply Descartes' rule of signs to find the nature of the roots of the equation, we need to determine the number of positive and negative roots.
1. Start by counting the sign changes in the polynomial equation:
- Write down all the coefficients of the equation, including zeros where there is no term.
- Look for the sign changes in the coefficients as you scan from left to right.
- Count the number of times the sign changes.
In our example equation: x^4 + 16x^2 + 7x - 11 = 0, there is only one sign change from positive to negative (from 7x to -11).
2. Next, to find the number of positive roots, substitute -x into the equation:
- Replace every instance of x with -x.
- Simplify the equation by combining like terms.
Replacing x with -x in our equation: (-x)^4 + 16(-x)^2 + 7(-x) - 11 = 0, will give us: x^4 + 16x^2 - 7x - 11 = 0.
Now, repeat step 1 for this new equation and count the number of sign changes. In this case, there are two sign changes: from positive to negative between 16x^2 and -7x, and from negative to positive between -7x and -11.
3. From the sign changes, we can determine the possible number of positive and negative roots:
- The number of positive roots is equal to the number of sign changes or less by an even number.
- The number of negative roots is equal to the number of sign changes or less by an odd number.
In our example, there is only one sign change and no even number of sign changes. Therefore, there is either one positive root or no positive roots.
There are two sign changes in the equation obtained by substituting -x. Since this is an odd number, there are either two negative roots or no negative roots.
So, based on Descartes' rule of signs, there is either one positive root or no positive roots and either two negative roots or no negative roots in the equation x^4 + 16x^2 + 7x - 11 = 0.