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.

Question

t(x) = 4x⁴ - 16x³ + 11x² + 4x - 3

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.

Upper Bound:

Lower Bound:

Answer 1

t(x) = 4x⁴ - 16x³ + 11x² + 4x - 3

Using the rational root theorem, the possible zeros are

{-1, 1, -3, 3, -3/4, 3/4, -3/2, 3/2, -1/2, 1/2, -1/4, 1/4}

To check for upper bounds, I'm going to plug in a positive number from the possible zeros using the Rational Root Theorem. Let's try -3/4.

 

 -3/4 |   4 -16 11 4 -3

         -3 57/4 -303/16 717/64

  -----------------------------------------------

*4  -19  101/4 -239/16 525/64*

*You see how the bottom numbers alternate from positive to negative (4, -19, 101/4, -239/16, 525/64)? That tells you that you have lower bound at x = -3/4. When you get a scenario like that when the numbers inside the * switch from positive to negative or vice versa, that's a lower bound. What this also tells you is that when finding zeros, they will never be lower than -3/4 since that's our lower bound, so we can already eliminate -3, -3/2 and -1 as possible zeros.

 

 

To check for upper bounds, I need to make sure that the numbers inside the * are ALL NON-NEGATIVE. Let's try x = 3.

 

 3 |  4 -16 11 4 -3

          12  -12 -3 3

   --------------------------------

      4  -4   -1  1 0

You see how the numbers AREN'T all non-negative? That means that 3 is NOT the upper bound. In this case, we need to go outside our possible zeros and try x=4. When we do that, we get:

4 |  4 -16 11 4 -3

16 0 44 192

--------------------------------

*4  0 11 48 189*

You see that when x = 4, my numbers inside the * are all non-negative. That means that the upper bound is at x = 4. This tells us that there will be no zeros that are greater than 4. So to summarize:

Upper Bound: 4

Lower bound: -3/4

This helps tremendously because we know that our zeros are between -3/4 and 4, so we can eliminate -3, -3/2, and -1.

Now to find the zeros: When you use Synthetic Division, It turns out that x = 1 is a zero. When the number in the * is a zero, that's a solution.

 

Synthetic Division:

 

  1 |   4 -16 11 4 -3

             4 -12 -1 3

   ------------------------

       4 -12 -1 3 *0*

 

This translates to 4x³ - 12x² - x + 3 = 0 

I can factor this out to (4x² - 1)(x - 3) = 0.

Using the differences of 2 squares method, I can factor 4x² - 1 into (2x - 1)(2x + 1), so 4x³ - 12x² - x + 3 factored out is (2x + 1) (2x - 1) (x - 3) = 0. Set each factor equal to zero to get -1/2, 1/2 and 3.

So my four zeros are -1/2, 1/2, 1 and 3.