upper bound is assumed to be x = 3
lower bound is assumed to be x = -4
the test for upper and lower bounds of the real zeroes can be found at the following link.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut39_zero2.htm
excerpt from that tutorial is shown below:
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Upper Bound
If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all positive numbers, then c is an upper bound to the real roots of the equation f(x) = 0.
Note that two things must occur for c to be an upper bound. One is c > 0 or positive. The other is that all the coefficients of the quotient as well as the remainder are positive.
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Lower Bound
If you divide a polynomial function f(x) by (x - c), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f(x) = 0. Special note that zeros can be either positive or negative.
Note that two things must occur for c to be a lower bound. One is c < 0 or negative. The other is that successive coefficients of the quotient and the remainder have alternating signs.
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if you don't know how to do synthetic division, then check out the following tutorial:
http://www.purplemath.com/modules/synthdiv2.htm
applying the upper bound test to 2x^4 - 8x + 3, you will get the following:
first of all you have to replace the missing degrees.
you also have to sort the equation in descending degree if it is not already sorted that way.
the equation of 2x^4 - 8x + 3 becomes:
2x^4 + 0x^3 + 0x^2 - 8x + 3
if c is equal to 3, then x-c is equal to x-3.
you will be dividing 2x^4 + 0x^3 + 0x^2 - 8x + 3 by (x-3) using synthetic division.
take the coefficients to form the dividend of the synthetic division.
you will get:
2 + 0 + 0 - 8 + 3
you will be dividing 3 into 2 + 0 + 0 - 8 + 3 using synthetic division.
the answer will be 2 + 6 + 18 + 46 with a remainder of 141.
this represents the quotient of 2x^3 + 6x^2 + 18x + 46 + 141/(x-3)
since all the coefficients are positive and the remainder is positive, then x = 3 is an upper bound of the real roots.
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the lower bound test will have x = -4
if c is equal to -4, then x - c is equal to x - (-4) which is equal to x + 4.
you will be dividing 2x^4 + 0x^3 + 0x^2 - 8x + 3 by (x+4) using synthetic division.
take the coefficients to form the dividend of the synthetic division.
you will get:
2 + 0 + 0 - 8 + 3
you will be dividing -4 into 2 + 0 + 0 - 8 + 3 using synthetic division.
the answer will be 2 - 8 + 32 - 136 with a remainder of 547.
this represents the quotient of 2x^3 - 8x^2 + 32x - 136 + 547/(x+4)
since the signs of the quotient and the remainder are alternating between positive and negative, then x = -4 is a lower bound of the real roots.
the details of the calculations are shown below:
the graph of the original equation of 2x^4 - 8x + 3 is shown below:
notice that the roots are not at -4 and 3.
they are between - 4 and 3.
in fact, the upper and lower bounds could have been more restrictive.
upper bound could easily have been 2.
lower bound could easily have been -1.