For the following function: `f(x)= x^6 -7x^4 -2x +7`
- Find the maximum number of real zeros, and
- Use Descartes's rule of signs to determine how many positive and how many negative zeros the function has. You do not need to find the zeros.
To answer this question, you must furst look at the fundamental theorem of algebra, which effectively states that a function has as many roots as the degree of the polynomial.
In this case, our degree is 6, so we'll have a maximal number of 6 real roots.
That wasn't too bad!
Now, let's use the Descartes' Rule of Signs to find the number of positive roots. To do this, we'll need to order the terms in terms of highest to lowest degree (which is already done for us!):
`f(x) = x^6-7x^4-2x+7`
Now, in this rule, we ignore coefficients that are equal to zero. So, all we're doing is looking to see when we have a positive coefficient followed by a negative coefficient and vice versa. In other words, we're looking for when there's a shift from + to - or - to + from left to right.
We can see two sign changes: between `x^6` and `-7x^4` and between `-2x` and `7`. Notice in the first case we are going from a coefficient of +1 to a coefficient of -7. In the second case, we again change sign from -2 to +7.
So, our number of sign changes is 2. Now, the rule of signs states that the number of roots will be either this number or this number minus some multiple of 2. For example, if there had been 20 sign changes, we could have 20, 18, 16, 14,...,4,2, or 0 roots. In our case, this indicates that we may have 2 or 0 positive roots.
To find the number of negative roots, we simply substitute -x for x in the function:
`f(x) = (-x)^6 -7(-x)^4 -2(-x) + 7`
To simplify, remember even powers of -x have the negatives cancel each other out. Odd powers don't. In other words, we look at terms of odd degrees, and just flip their sign!
` ` `f(x) = x^6-7x^4 + 2x + 7`
Now, we count sign changes again, and again, we have two!
Between `x^6` and `-7x^4` (+1 to -7) and between `-7x^4` and `2x`(-7 to +2).
So, we again have two sign changes, and the rule is the same, where the number of roots is either this number or it minus some multiple of 2.
Therefore, we have 2 or 0 negative roots!
I hope that helps! This rule does appear to tell us we can have a maximum of 4 real roots (because clearly, there are no roots at x=0). However, without having checked this first, we had no reason to believe that this function had any less than 6 real roots!