List the possible rational roots of each equation. Then determine the rational roots.
6x^4+35x^3-x^2-7x-1
The answer in the back of the book is:
-1/3 and 1/2
I know that I'm supposed to use Descartes rule of signs. I found that there was 1 positive zero and 3 or 1 negative zeroes.
I'm a bit confused on how you know exactly how many of each zeroes you are supposed to have. I'm assuming you know by looking at the degree and determining which adds up.
Also, how come the book only states 2 answers?
I used an online calculator that says there are 2 other answers which are:
-3 - 2(2)^(1/2)
and
2(2)^(1/2) - 3
Much help appreciated
To determine the possible rational roots of a polynomial equation, you can use the Rational Root Theorem. The Rational Root Theorem states that if a polynomial equation has a rational root (a root that can be expressed as a fraction), it must be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the polynomial equation is 6x^4 + 35x^3 - x^2 - 7x - 1.
The constant term is -1, and its factors are ±1.
The leading coefficient is 6, and its factors are ±1, ±2, ±3, and ±6.
So the possible rational roots can be calculated by taking all the combinations of pairs consisting of a factor of the constant term and a factor of the leading coefficient.
Possible pairs of the form p/q are:
±1/1, ±1/2, ±1/3, ±1/6,
±1/1, ±2/2, ±2/3, ±2/6,
±1/1, ±3/2, ±3/3, ±3/6,
±1/1, ±6/2, ±6/3, ±6/6.
Simplifying these fractions:
±1, ±1/2, ±1/3, ±1/6,
±1, ±1, ±2/3, ±1/3,
±1, ±3/2, ±1, ±1/2,
±1, ±3, ±2, ±1.
Now we need to test each of these possible rational roots to see if any of them are actual roots of the polynomial equation.
Using synthetic division or polynomial long division, we can test each possible root by dividing the polynomial equation by the trial root and see if the remainder is zero.
By testing each possible root, you will find that the rational roots for this equation are -1/3 and 1/2.
Now, regarding your confusion about Descartes' Rule of Signs, it is a theorem that provides information about the number of positive and negative real roots of a polynomial equation. It states that the number of positive real roots of a polynomial equation is either equal to the number of sign changes in the coefficients or less than it by an even number. Similarly, the number of negative real roots is equal to the number of sign changes in the coefficients or less than it by an even number.
In this case, Descartes' Rule of Signs tells us that there is 1 positive root and either 3 or 1 negative roots. It doesn't provide the actual values of these roots, just the number of them.
As for the reason the book states only two answers, it is because it specifically asks for the rational roots. The other two answers you found using an online calculator are the complex (non-real) roots of the equation. The book might not have included these roots because it was focusing only on the rational (real or integer) roots.
I hope this explanation helps! Let me know if you have any further questions.