Question 1182855
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Use Descartes' Rule of Signs.<br>
(1) Put the polynomial in standard form -- that is, with decreasing powers of x.
(2) The possible numbers of positive real roots are the number of sign changes in the terms of the polynomial, minus any multiple of 2.
(3) The possible numbers of negative real roots are the number of sign changes in the terms of the polynomial when x is replaced by -x, minus any multiple of 2.<br>
For the polynomial in your problem....<br>
3x^4+4x^3+x-1
The signs of the terms are +, +, +, -
There is 1 sign change
The only possible number of positive real roots is 1<br>
Replace x with -x in the polynomial; note that the signs of the coefficients of the even degree terms stay the same while the signs of the coefficients of the odd degree terms change.
3x^4-4x^3-x-1
The signs are +, -, -, -
There is 1 sign change
The only possible number of negative real roots is 1<br>
The polynomial is degree 4, so it has 4 roots.  There is only 1 positive real root and only 1 negative real root, so<br>
ANSWER: D. The polynomial has one positive real root, one negative real root, and one pair of complex roots.<br>
For a more interesting example of applying Descartes' Rule of Signs, consider this random polynomial:<br>
4x^8+3x^7-5x^6-3x^5-2x^3+x^2+3x-1<br>
The signs of the terms are +, +, -, -, -, +, +, -
There are 3 sign changes
The number of positive real roots is either 3 or 1<br>
Replace x with -x:
4x^8-3x^7-5x^6+3x^5+2x^3+x^2-3x-1
The signs of the terms are +, -, -, +, +, +, -, -
There are 3 sign changes
The number of negative real roots is either 3 or 1<br>
The nature of the roots is...
either 1 or 3 positive real roots
either 1 or 3 negative real roots<br>
The total number of real roots is then either 2, or 4, or 6; Since there are 8 roots, there are...
either 6, or 4, or 2 complex roots<br>