Consider the following polynomial D(x) = x^4+6x^3+7x^2-6x-8

Question

Use synthetic division to identify integer bounds of the real zeros. Find the least upper bound and the greatest lower bound guaranteed by the upper and lower bounds of zeros theorem

Answer 1

1) The Upper Bound Theorem

An "Upper Bound" means there will be no zeros greater than that bound. The Upper Bound Theorem says that if you divide a polynomial by (x - c), i.e., you use synthetic division using "c", then "c" is an upper bound if the results of the synthetic division yields all positive numbers (including the remainder). Note: You can count zeros as either positive or negative.

2) The Lower Bound Theorem

A "Lower Bound" means there will be no zeros less than that bound. The Lower Bound Theorem says that if you divide a polynomial by (x - c), i.e., you use synthetic division using "c", then "c" is an lower bound if the results of the synthetic division yields alternating positive/negative numbers (including the remainder). Note: You can count zeros as either positive or negative.

Let's start trying to find the Lower Bound, The Synthetic Division layout will look like this:

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c | 1 6 7 -6 -8

Since we are looking for alternating +/- numbers, we can see that for the second number , "6", we must add a number > -6 in order to get the result to be negative (or zero). So, let's try -6:

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-6 | 1 6 7 -6 -8

-6 0 -42 288

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1 0 7 -48 280

This works, so -6 is our Lower Bound since any other number bigger will not result in a negative for the second number).

To find the upper bound, let's arbitrarily try 2:

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2 | 1 6 7 -6 -8

2 16 46 80

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1 8 23 40 72

This works but perhaps a smaller number will work. Let's try 1:

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1 | 1 6 7 -6 -8

1 7 14 8

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1 7 14 8 0

This also works and it is also a zero

So the lower bound is -6 and the upper bound is 1. All zeros will be between these numbers.