# 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

embizze | High School Teacher | (Level 1) Educator Emeritus

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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`