Please help. Use Descartes' rule of sign to describe the roots of h(x)=4x^4-5x^3+2x^2-x+5
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These are +4, -5, +2, - 1, + 5. Next focus on the sign changes as you go along from left to right. We are going to count them. Bracket them as follows: (+4, -5)→(-5, +2)→(+2, -1)→(-1, + 5). So as you go along the sign has changed 4 times. Descartes' rule of signs tells us that the number of positive roots is either 4 or 4 less an even number, I.e., 4 0r 4 - 2 = 2 or 4 - 4 = 0. So there are 0, 2 or 4 +ve roots. IT DOES NOT TELL US WHICH OF 0, 2, 4 IT IS. Now for the -ve roots. These are just the number of +ve roots of h(-x), which is 4x⁴+ 5x³ + 2x² +x +5. There are no changes of sign here, so h(-x) has 0 +ve roots, i.e., h(x) has 0 -ve roots. Thus, if none of the roots are complex all its 4 roots must be +ve, but we don't know this. So all we can know from the rule of signs is that the polynomial has no -ve roots and 0, 2 or 4 +ve ones.
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4x4−5x3+2x2−x=−5
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