confused about polynomial!!PLEASE HELP! (1 Viewer)

coeyz · Feb 27, 2009

1. Prove (x^4)/4 + (x^3)/3 + (x^2)/2 + x + C has no real zeros if C > 7/12

Teacher told us to find the min.tp, and show that the y value of the tp is < 0,
then there will be no real zero.
but how about max tp? if a say max tp with is < 0, is it also no real zeros?

another question, if C < 7/12, how many real roots are there?

2. determine the number of real zeros of the polynomial
y= 6x^5 - 10x^3 + 15x^2 + 300x + 30

finding number of real zeros again!!
wht should i do?

some people said if theres real zero, then P'(x) > 0
but why??? i dont get it =(

tommykins · Feb 27, 2009

1.
$y = \frac{x^{4}}{4} + \frac{x^{3}}{3} + \frac{x^{2}}{2} + x + C\\ \frac{dy}{dx} = x^3 + x^2 + x + 1 = x^2(x+1) + x+1 = (x+1)(x^2+1)\\ \text{Turning point at x = -1}\\ \frac{d^2y}{dx^2} = 3x^2 + 2x + 1 \rightarrow \text{when x = -1} \rightarrow 2 \therefore \text{concave up minimum turning point.}\\ y = \frac{1}{4} - \frac{1}{3} + \frac{1}{2} - 1 + C \rightarrow \text{When x = -1}\\ y = \frac{-7}{12} + C \therefore \text{for y to have no real roots,} y > 0 \\\frac{-7}{12} + C > 0 \Rightarrow C > \frac{7}{12}$

There should be 2 real roots if C < 7/12

2. If you notice the polynomial, differentiating gives a coefficient of factor 30. Do so and use the same principle above and apply it.

And no, if P'(x) > 0 for all x, then the graph is monotonically increasing and hence has a MAXIMUM of one real root.

confused about polynomial!!PLEASE HELP! (1 Viewer)

coeyz

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tommykins

i am number -e^i*pi

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