# use the syntheic division to verify the upper and /or lower bounds of the zeros of f. f(x)=2x^2-8x+3 upperbound:x=3; lowerbound x=-4

### 1 Answer | Add Yours

Using synthetic division, we are asked to find the upper and lower bounds on the roots of `f(x)=2x^2-8x+3` :

(1) If you divide `f(x)` by `(x-k)` and the coefficients of the quotient and the remainder are all nonnegative, there are no real zeros greater than k. Using synthetic division, the bottom row will be all positive.

**3 is not an upper bound for the given function:**

3| 2 -8 3

---------

2 -2 -3

This can be confirmed since one of the real roots of f(x) is `2+sqrt(10)/2~~3.18`

However, 4 is an upper bound since:

4| 2 -8 3

----------

2 0 3

(2) If you divide f(x) by (x-k) and the coefficients of the quotient and remainder alternate in sign, then k is a lower bound.

**Here, -4 is certainly a lower bound:**

-4 | 2 -8 3

----------

2 -16 67

-1 is also a lower bound:

-1 | 2 -8 3

----------

2 -10 13

1 is not a lower bound:

1 | 2 -8 3

---------

2 -6 -3

This can be verified as the other real root of f(x) is `2-sqrt(10)/2~~.419`