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.

Question

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.

 

F(x)=x^3-5x^2-6x+30

Answer 1

Applying the rational root theorem, the possible candidates are

{+or- 1, +or- 2, +or- 3, +or- 5, +or- 6, +or- 10, +or- 15, +or- 30}

 

It turns out that x=5 is a zero.

 

Synthethic Division:

 

    5 |     1 -5   -6    30

                 5    0   -30

      ------------------------

              1 0    -6   0

 

x^2 - 6 = 0

 

x = +or- sqrt(6)

 

 

Using synthetic division on x=7 to check for upper bounds:

 

  7 |      1 -5   -6   30

                7    14   56

    -------------------------

            1  2    8   86

 

So 7 is an upper bound

 

 

Checking lower bounds with x=-3

 

 -3 |    1  -5    -6   30

              -3    24  -54

     ------------------------

          1  -8   18  -24

 

The signs of these coefficients alternate,

 so -3 is a lower bound

 

Since it is a cubic polynomial, there are only 3 zeros.

 

They are bounded as:

 

  -3 < -sqrt(6) < sqrt(6) < 5 < 7

 

 

             

 

Answer 2

In order to use this theorem, the leading coefficient of the polynomial must be positive and must have real coefficients.

 

If you divide the coefficients of F(x) by a possible positive root c, and the quotient & remainder have positive signs, then the root is an upper bound.

 

If you divide the coefficients of F(x) by a possible negative root c, and the quotient & remainder have alternating signs, then the root is a lower bound.

 

 

Using the rational root theorem, lets test c=5.

 

5 | 1        -5           -6           30

 

               5            0           -30

_____________________________

     1        0            -6           0

 

 

c=5 is a root.  But the signs are not all positive.  We will try a greater root.

 

 

c=15

 

 

15 | 1         -5          -6           30

 

                  15        150         

________________________________

       1         10       144        positive #

 

 

15 is an upper bound.

 

 

 

Now we look at negative roots.

 

Test c=-3

 

-3 | 1     -5        -6        30

 

             -3       24       -66

_______________________

      1     -8      22         -36

 

 

The signs alternate.

 

 

Your zeros are in the interval (-3, 15).